What is the area of the region bounded by the lines x=1 and y=0 and the curve y= x(e^x^2)?
A. 1-e units squared
B. e-1 units squared
C. (e-1)/2 units squared
D. (1-e)/2 units squared
or E. e/2 units squared
if you can show me how to get that please
To find the area of the region bounded by the lines x=1, y=0, and the curve y= x(e^x^2), we can use the definite integral.
Step 1: Determine the intersection points of the curve and the lines.
- To find the intersection points, set the curve equation equal to the line equations: x(e^x^2) = 1 and x = 1.
- Setting x = 1 in the curve equation gives you y = 1(e^1^2) = e.
- Therefore, the curve intersects the line y=0 at x = 1 and y = 0, and it intersects the line x =1 at (1, e).
Step 2: Find the integral bounds.
- The area we want to calculate is between the curve and the x-axis, bounded by x=1 and the intersection point (1, e).
- So, our integral bounds will be from x = 1 to x = 1.
Step 3: Set up the definite integral to find the area.
- The area can be calculated using the definite integral of the function y= x(e^x^2) from x=1 to x=1.
- The integral will be ∫[1, 1] x(e^x^2) dx.
Step 4: Evaluate the definite integral.
- To evaluate the integral, we can use integration by substitution.
- Let u = x^2. Then, du = 2x dx.
- Rearranging the terms, we have x dx = (1/2) du.
- Substituting into the integral, it becomes ∫[1, 1] (1/2)e^u du.
- Since the bounds are the same, the integral becomes 0.
Therefore, the area bounded by the lines x=1, y=0, and the curve y= x(e^x^2) is 0.
Based on the answer choices given, none of them match the correct answer.