Test: Double And Triple Integrals - 1 - Mathematics MCQ

To derive the volume of an ellipsoid given by the equation

we can calculate the volume by setting up a triple integral in spherical coordinates. However, there is a standard approach to find this without an explicit integral due to the symmetry of the ellipsoid.

Step-by-Step Calculation:

1. Transform to Spherical Coordinates: To simplify the volume calculation, consider an ellipsoid centered at the origin. We can relate this ellipsoid to a unit sphere by performing a scaling transformation in the x, y, and z directions:
   
   With these substitutions, the ellipsoid equation becomes that of a unit sphere:
   

2. Volume of the Unit Sphere: The volume of a unit sphere (with radius 1) is well known and given by:
   

3. Scaling Factor: Now, to scale this volume to our ellipsoid, we account for the scaling transformations applied to each axis:

• In the x-direction, the scaling factor is a,
• In the y-direction, the scaling factor is b,

So, the volume of the ellipsoid becomes the volume of the unit sphere multiplied by a × b × c:

Final Answer: Thus, the volume of the ellipsoid is:

This confirms that option B is correct.

To find the area bounded by the curve y = ψ(x), the x-axis, and the vertical lines x = l and x = m (where l < m), we can set up an integral to calculate this area.

The area A can be calculated as the integral of y = ψ(x) with respect to x, over the interval from x = l to x = m.

Thus, the area A is given by:

A = ∫_l^m ∫₀^ψ(x) dy dx.

This is because:

• For each fixed x in the interval [l, m], y ranges from 0 (the x-axis) up to y = ψ(x).
• The outer integral integrates this area along x from l to m.

In this double integral setup:

A = ∫_l^m ( ∫₀^ψ(x) dy ) dx.

Evaluating the inner integral:

∫₀^ψ(x) dy = ψ(x).

So the expression simplifies to:

A = ∫_l^m ψ(x) dx,

which matches option B:

∫_l^m ψ(x) dx.

Conclusion:

The correct answer is:

B: ∫_l^m ψ(x) dx.

This confirms that option B is indeed correct.

The volume integral in spherical coordinates is given by:

V = ∫₀^2π ∫₀^π/3 ∫₀¹ r² sin(φ) dr dφ dθ.

Step 1: Integrate with respect to r

The inner integral is with respect to r from 0 to 1:

∫₀¹ r² dr = [ r³/3 ]₀¹ = 1/3.

Substituting this result, we now have:

V = ∫₀^2π ∫₀^π/3 (1/3) sin(φ) dφ dθ.

Step 2: Integrate with respect to φ

Now we integrate with respect to φ from 0 to π/3:

∫₀^π/3 sin(φ) dφ = [ -cos(φ) ]₀^π/3 = -cos(π/3) + cos(0).

Since cos(π/3) = 1/2 and cos(0) = 1, this becomes:

= -1/2 + 1 = 1/2.

Substituting this result, we get:

V = ∫₀^2π (1/3) * (1/2) dθ = (1/6) ∫₀^2π dθ.

Step 3: Integrate with respect to θ

Now we integrate with respect to θ from 0 to 2π:

∫₀^2π dθ = 2π.

So, we have:

V = (1/6) * 2π = π/3.

Conclusion:

The value of the integral is:

A: π/3.

To evaluate the integral:

∫₁^{log 8} ∫₀^{log y} e^x+y dx dy,

we solve it by integrating with respect to x first, followed by y.

Step 1: Integrate with respect to x

The inner integral is:

∫₀^{log y} e^x+y dx.

Since e^x+y = e^x * e^y, we can rewrite this as:

= e^y ∫₀^{log y} e^x dx.

Integrating e^x with respect to x from 0 to log y:

∫₀^{log y} e^x dx = [ e^x ]₀^{log y} = y - 1.

Therefore, the inner integral becomes:

∫₀^{log y} e^x+y dx = e^y (y - 1).

Step 2: Integrate with respect to y

Now we substitute back into the outer integral:

∫₁^{log 8} e^y (y - 1) dy = ∫₁^{log 8} (y e^y - e^y) dy.

This expands to:

∫₁^{log 8} y e^y dy - ∫₁^{log 8} e^y dy.

Integral 1: ∫₁^{log 8} y e^y dy

To solve ∫ y e^y dy, we use integration by parts with:

- u = y and dv = e^y dy,
- then du = dy and v = e^y.

Using integration by parts:

∫ y e^y dy = y e^y - ∫ e^y dy = y e^y - e^y.

Evaluating this from 1 to log 8:

[ y e^y - e^y ]₁^{log 8} = ( (log 8) * 8 - 8 ) - ( e - e ) = 8 log 8 - 8.

Integral 2: ∫₁^{log 8} e^y dy

This integral is straightforward:

∫₁^{log 8} e^y dy = [ e^y ]₁^{log 8} = 8 - e.

Final Calculation

Putting it all together:

∫₁^{log 8} ∫₀^{log y} e^x+y dx dy = (8 log 8 - 8) - (8 - e).

Simplifying:

= 8 log 8 - 8 - 8 + e = 8 log 8 - 16 + e.

Conclusion:

The correct answer is: C: 8 log 8 - 16 + e.

To solve this problem, we need to evaluate the integral:

∫₀¹ ∫₀¹ |cos(x + y)| dx dy

using the transformation x + y = u and y = v.

Step 1: Determine the Transformation and Jacobian

The transformation is given by:

• u = x + y
• v = y

To express x and y in terms of u and v:

• From u = x + y, we can solve for x as x = u - v.
• Thus, x = u - v and y = v.

Now, we need to calculate the Jacobian J of the transformation:

Calculating each partial derivative:

• ∂x/∂u = 1
• ∂x/∂v = -1
• ∂y/∂u = 0
• ∂y/∂v = 1

So, the Jacobian is:

J = | 1   -1 | = (1)(1) - (-1)(0) = 1.
     | 0    1 |

Thus, the Jacobian J = 1.

Step 2: Set Up the Integral in Terms of u and v

Since J = 1, the integral becomes:

∫₀¹ ∫₀¹ |cos(u)| du dv.

Step 3: Evaluate the Integral

We can separate the integrals because |cos(u)| depends only on u:

∫₀¹ ∫₀¹ |cos(u)| du dv = ∫₀¹ ( ∫₀¹ |cos(u)| du ) dv.

First, we evaluate the inner integral with respect to u:

∫₀¹ |cos(u)| du.

Since cos(u) is positive over the interval [0, 1], we have:

∫₀¹ cos(u) du = [sin(u)]₀¹ = sin(1) - sin(0) = sin(1).

Now, substitute this result back:

∫₀¹ ∫₀¹ |cos(u)| du dv = ∫₀¹ sin(1) dv = sin(1) * [v]₀¹ = sin(1).

Conclusion:

The value of the integral is approximately: B: 1.

We need to find the area bounded by the parabola y² = 4ax and the straight line x + y = 3a.

Step 1: Rewrite the Equations

The given equations are:

• Parabola: y² = 4ax
• Line: x + y = 3a, which can be rewritten as y = 3a - x

Step 2: Find Points of Intersection

To find the intersection points, substitute y = 3a - x into y² = 4ax:

(3a - x)² = 4ax.

Expanding and simplifying:

9a² - 6ax + x² = 4ax,

x² - 10ax + 9a² = 0.

This is a quadratic equation in x:

(x - 5a)² = 0.

So, x = 5a. Substituting x = 5a back into y = 3a - x:

y = 3a - 5a = -2a.

Thus, the point of intersection is (5a, -2a).

Step 3: Set Up the Integral for the Area

To find the area bounded by the parabola and the line, we integrate horizontally from x = 0 to x = 5a, finding the difference between the y-values of the line and the parabola at each x.

The area A is given by:

A = ∫₀^5a ((3a - x) - √(4ax)) dx.

Step 4: Evaluate the Integral

We will split the integral into two parts:

1. Integral of (3a - x):

2. Integral of √(4ax):

Rewrite √(4ax) = 2√(ax), so we have:

Conclusion:

The correct answer, after calculating both integrals, is:

D: 10a² / 3.

The equation of line in intercept form is given by

We assume x + 2 = t² dx = 2t dt

To evaluate the double integral:

∫_θ₁^θ₂ ∫_r₁^r₂ f(r, θ) r dr dθ

over the region A bounded by r = r₁, r = r₂, and the straight lines θ = θ₁ and θ = θ₂, we analyze the integration process.

Explanation of the Limits of Integration

1. First Integration (Inner Integral)

We integrate with respect to r (the radial distance) from r = r₁ to r = r₂.

• During this integration, θ is treated as a constant since it is independent of r.

Therefore, we evaluate r between the limits r = r₁ and r = r₂, treating θ as a constant. This matches option A.

2. Second Integration (Outer Integral)

After integrating with respect to r, we then integrate with respect to θ from θ = θ₁ to θ = θ₂.

• During this integration, r is treated as a constant since it has already been integrated out in the previous step.

Thus, we evaluate θ between the limits θ = θ₁ and θ = θ₂, treating r as a constant. This matches option B.

Conclusion

Since both statements A and B are correct, the answer is: C: a) and b) both.

To convert a point from Cartesian coordinates to spherical coordinates, use equations 
 

r² = x² + y² + z² , tanθ = y/x,   φ = arccos(z/√(x²+y²+z²) )

where V is region bounded by the plane 2x + y + z = 4 x = 0, y = 0, z = 0


From (1) & (2), we have
⇒ λ(x) = 4 – 2x.
µ(x, y) = 4 – 2x – y
⇒ λ – µ = 4 – 2x – 4 + 2x + y = y.

To find the area bounded by the curves y² = x³ and x² = y³, let's go through the following steps.

Step 1: Find Points of Intersection

Set y² = x³ equal to x² = y³ to find their intersection points.

1. From y² = x³, we get y = x^3/2 (considering only the positive values since we are looking for the bounded area in the first quadrant).
2. Substitute y = x^3/2 into x² = y³:

x² = (x^3/2)³

This simplifies to:

x² = x^9/2

Divide both sides by x^3/2 (assuming x ≠ 0):

x = 1

Thus, the points of intersection in the first quadrant are (1, 1) and (0, 0).

Step 2: Set Up the Integral

To find the area bounded by these curves from x = 0 to x = 1:

• For y² = x³, solve for y: y = x^3/2.
• For x² = y³, solve for y: y = x^2/3.

The area A between these curves from x = 0 to x = 1 is given by:

A = ∫₀¹ (x^2/3 - x^3/2) dx.

Step 3: Evaluate the Integral

Separate the integral:

A = ∫₀¹ x^2/3 dx - ∫₀¹ x^3/2 dx.

1. Integral of x^2/3

∫₀¹ x^2/3 dx = [ x^5/3 / (5/3) ]₀¹ = 3/5.

2. Integral of x^3/2

∫₀¹ x^3/2 dx = [ x^5/2 / (5/2) ]₀¹ = 2/5.

Now, substitute these values back:

A = 3/5 - 2/5 = 1/5.

Conclusion

The area bounded by the curves y² = x³ and x² = y³ is:

C: 1/5.

To solve the integral

we'll evaluate it step by step, starting with the innermost integral and moving outward.

Step 1: Evaluate the Inner Integral with Respect to z

Step 2: Substitute and Evaluate the Middle Integral with Respect to y

Now, integrate term by term with respect to y:

Step 3: Evaluate the Outer Integral with Respect to x

Conclusion: The value of the integral is: 7/8.

So, the correct answer is: D: 7/8

To evaluate the integral:

∫₀^a ∫_y^a (x / (x² + y²)) dx dy

by changing the order of integration, we need to determine the region of integration in the x-y plane and then rewrite the integral accordingly.

Step 1: Determine the Region of Integration

The given limits indicate that:

• y ranges from 0 to a,
• For a fixed y, x ranges from y to a.

In the x-y plane, this describes the triangular region bounded by:

• y = 0,
• x = a,
• x = y.

Step 2: Change the Order of Integration

To change the order of integration, we need to describe the region in terms of x first:

• x ranges from 0 to a,
• For a fixed x, y ranges from 0 to x (since y ≤ x within this region).

Thus, we can rewrite the integral as:

Step 3: Evaluate the Inner Integral with Respect to y

Now, we evaluate the inner integral:

Since x is treated as a constant in the inner integral, we can factor it out:

Integrate with respect to y:

Since arctan(1) = π/4 and arctan(0) = 0, this becomes:

= (1/x) * (π/4) = π / (4x).

Step 4: Substitute Back and Integrate with Respect to x

Now our integral becomes:

This simplifies to:

Conclusion:

The value of the integral is:

B: πa / 4.

Step 1: Calculate Partial Derivatives

Given:

We find the partial derivatives as follows:

Step 2: Set Up the Jacobian Determinant

Step 3: Calculate the Determinant

The absolute value of J (since we are only concerned with the magnitude in the context of integration) is:

Conclusion: The value of ϕ(u, v) is: a. 2u/v

Here
Let u = x/a, v = y/b , w = z/c
Then dx = a du, dy = b dv, dz = c dw
So, Required volume
V = abc du dv dw
where u + v + w ≤ 1, u ,v ,w ≥ 0
Thus 

Hence, the correct answer is (d)

I = dxdy

Putting x² + y² = r², we get
Limits of  r = 0 to ∞
and 0 = θ to π/2
Now, Putting r² = t, we get

Step 1: Simplify by Changing Variables

To simplify this integral, note that we can use the property of separable integrals by changing variables. However, for clarity, let’s proceed with evaluating the inner integral directly.

Step 2: Evaluate the Inner Integral with Respect to x

Step 3: Integrate with Respect to y

Conclusion: The value of the integral is: 2.
So, the correct answer is: D: 2.

Step 1: Rewrite the Integrand

Notice that we can rewrite the expression as:

This allows us to separate the integral into two parts, one in terms of x and the other in terms of y.

Step 2: Separate the Integral

We can write the integral as:

Step 3: Evaluate the Integral with Respect to y

Consider the integral:

Step 4: Evaluate the Integral with Respect to x

Now consider the integral:

Step 5: Combine the Results

Now, we combine both parts:

Conclusion: The value of the integral is: (ye^y - e^y)(xe^x - e^x)
So, the correct answer is Option B.