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A triangle ABC consists of vertex points A (0,0) B(1,0) and C(0,1). The value of the integral ∫∫ 2x dxdy over the triangle is
- a)1
- b)1/3
- c)1/8
- d)1/9
Correct answer is option 'B'. Can you explain this answer?
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A triangle ABC consists of vertex points A (0,0) B(1,0) and C(0,1). Th...
To find the value of the integral, we need to set up the integral using the given triangle.
First, let's find the equation of the line AB. The slope of AB is (0-0)/(1-0) = 0, and since the y-coordinate of both points A and B is 0, the equation of the line AB is y = 0.
Next, let's find the equation of the line AC. The slope of AC is (1-0)/(0-0) = undefined, and since the x-coordinate of both points A and C is 0, the equation of the line AC is x = 0.
Now, we can set up the integral using the equation of the line AB and the equation of the line AC.
Since the triangle ABC is bounded by the x-axis, the y-axis, and the line AB, we can set up the integral as follows:
∫[y = 0 to y = 1] ∫[x = 0 to x = y] dx dy
Integrating the inner integral with respect to x, we have:
∫[y = 0 to y = 1] [x] [x = 0 to x = y] dy
= ∫[y = 0 to y = 1] y^2 dy
Integrating the outer integral with respect to y, we have:
∫[y = 0 to y = 1] (1/3) y^3 [y = 0 to y = 1]
= (1/3) [1^3 - 0^3]
= 1/3
Therefore, the value of the integral is 1/3.
First, let's find the equation of the line AB. The slope of AB is (0-0)/(1-0) = 0, and since the y-coordinate of both points A and B is 0, the equation of the line AB is y = 0.
Next, let's find the equation of the line AC. The slope of AC is (1-0)/(0-0) = undefined, and since the x-coordinate of both points A and C is 0, the equation of the line AC is x = 0.
Now, we can set up the integral using the equation of the line AB and the equation of the line AC.
Since the triangle ABC is bounded by the x-axis, the y-axis, and the line AB, we can set up the integral as follows:
∫[y = 0 to y = 1] ∫[x = 0 to x = y] dx dy
Integrating the inner integral with respect to x, we have:
∫[y = 0 to y = 1] [x] [x = 0 to x = y] dy
= ∫[y = 0 to y = 1] y^2 dy
Integrating the outer integral with respect to y, we have:
∫[y = 0 to y = 1] (1/3) y^3 [y = 0 to y = 1]
= (1/3) [1^3 - 0^3]
= 1/3
Therefore, the value of the integral is 1/3.
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A triangle ABC consists of vertex points A (0,0) B(1,0) and C(0,1). Th...
The equation of the line AB is
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