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Area of the region bounded by the curves y = ex, x = a , x = b and the x- axis is given by
- a)eb−ea
- b)eb−a
- c)eb+ea
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
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Area of the region bounded by the curves y =ex, x = a , x = b and the ...
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Area of the region bounded by the curves y =ex, x = a , x = b and the ...
To find the area of the region bounded by the curves y = ex, x = a, x = b, and the x-axis, we can use integration.
Step 1: Determine the limits of integration
The region is bounded by the vertical lines x = a and x = b, so the limits of integration for x will be from a to b.
Step 2: Set up the integral
The area of the region can be found by integrating the function y = ex with respect to x over the given limits of integration.
The integral for the area is given by:
∫(ex) dx from a to b
Step 3: Evaluate the integral
Integrating the function ex with respect to x gives us ex.
So, the integral becomes:
∫(ex) dx = ex
Evaluating this integral from a to b gives us:
ex evaluated from a to b = eb - ea
Step 4: Subtracting the areas below the x-axis
Since the region is bounded by the x-axis, we need to consider the areas below the x-axis as negative.
If there are any values of x within the interval (a, b) where ex is negative, we need to subtract the absolute value of the integral of ex in that range. However, since the function ex is always positive, we don't need to consider this step.
Step 5: Finalize the answer
The area of the region bounded by the curves y = ex, x = a, x = b, and the x-axis is given by:
Area = eb - ea
Therefore, the correct answer is option A) eb - ea.
Step 1: Determine the limits of integration
The region is bounded by the vertical lines x = a and x = b, so the limits of integration for x will be from a to b.
Step 2: Set up the integral
The area of the region can be found by integrating the function y = ex with respect to x over the given limits of integration.
The integral for the area is given by:
∫(ex) dx from a to b
Step 3: Evaluate the integral
Integrating the function ex with respect to x gives us ex.
So, the integral becomes:
∫(ex) dx = ex
Evaluating this integral from a to b gives us:
ex evaluated from a to b = eb - ea
Step 4: Subtracting the areas below the x-axis
Since the region is bounded by the x-axis, we need to consider the areas below the x-axis as negative.
If there are any values of x within the interval (a, b) where ex is negative, we need to subtract the absolute value of the integral of ex in that range. However, since the function ex is always positive, we don't need to consider this step.
Step 5: Finalize the answer
The area of the region bounded by the curves y = ex, x = a, x = b, and the x-axis is given by:
Area = eb - ea
Therefore, the correct answer is option A) eb - ea.
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Area of the region bounded by the curves y =ex, x = a , x = b and the x- axis is given bya)eb−eab)eb−ac)eb+ead)none of theseCorrect answer is option 'A'. Can you explain this answer?
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