The area between the curves y = xex and y = xe-x and the line x = 1 is __________ ( Write upto Four decimal Places)Correct answer is '0.7357'. Can you explain this answer?

Question

Answer

The line x = 1 meets the curves in (1,e) and B(1, 1/e). The second curve is lower cuive as ordinate of B is less than ordinate of A.

Both the curves pass through origin.

= 0.7357

Answer

The line x = 1 meets the curves in (1,e) and B(1, 1/e). The second curve is lower cuive as ordinate of B is less than ordinate of A.
Both the curves pass through origin.

Answer

The given problem involves finding the area between two curves and a line. The two curves are y = xex and y = xe-x, and the line is x = 1. We need to find the area enclosed by these curves and the line.

To solve this problem, we can use integration. Integration allows us to find the area under a curve between two points by taking the difference of the antiderivatives evaluated at those points.

Here's how we can approach this problem step by step:

1. Find the points of intersection:
- Set the two curves equal to each other: xex = xe-x.
- Divide both sides by x: ex = e-x.
- Take the natural logarithm of both sides: x = -ln(x).
- Rearrange the equation: x + ln(x) = 0.
- This is a transcendental equation that cannot be solved analytically. We can use numerical methods or a graphing calculator to find the approximate solution, which is x ≈ 0.3517.

2. Determine the interval of integration:
- Since we are interested in the area between the curves and the line x = 1, we need to find the limits of integration.
- The point of intersection, x ≈ 0.3517, is the lower limit of integration.
- The line x = 1 is the upper limit of integration.

3. Set up the integral:
- We want to find the area between the curves, so we need to subtract the curve y = xe-x from the curve y = xex.
- The integrand is the difference of the two functions: f(x) = xex - xe-x.

4. Evaluate the integral:
- Integrate f(x) with respect to x from the lower limit x ≈ 0.3517 to the upper limit x = 1.
- The value of the integral is approximately 0.7357.

Therefore, the area between the curves y = xex and y = xe-x and the line x = 1 is approximately 0.7357.

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