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
You need to specify both a lower and an upper limit for the x integration. You only specified one.
The indefinite integral of that function, which you need to solve the problenm is (1/2)e^x^2. The value of that function at x = 1 is e/2. The value at x = 0 is 1/2. If the lower limit of x integration is 0, the answer is (e-1)/2.
"units squared' refers to the area dimension; it does not mean you are supposed to square the number
To find the area of the region bounded by the lines x = 1, y = 0, and the curve y = x(e^x^2), you need to integrate the curve between the given x-values and subtract the area under the x-axis. Here's how you can do that:
Step 1: Determine the x-values at which the curve intersects the x-axis (y = 0). Set y = 0 in the equation y = x(e^x^2) and solve for x:
0 = x(e^x^2)
Since e^x^2 is always positive, this equation is satisfied when x = 0. Thus, the curve intersects the x-axis only at x = 0.
Step 2: Integrate the curve between the x-values of the boundaries (x = 0 to x = 1) to find the area above the x-axis:
A = ∫[0 to 1] x(e^x^2) dx
To integrate this function, use u-substitution. Set u = x^2 and du = 2x dx:
A = ∫[0 to 1] (1/2)e^u du (substituting x^2 for u)
Now integrate using the limits of integration [0 to 1]:
A = [(1/2)e^u] [0 to 1]
A = (1/2)e^1 -(1/2)e^0
A = (1/2)e - (1/2)
Step 3: Subtract the area under the x-axis. Since y = 0 lies on the x-axis, the area below the x-axis is equal to the absolute value of the integral of the curve between the same x-values:
A_under = ∫[0 to 1] |x(e^x^2)| dx
Using u-substitution as before, A_under can be written as:
A_under = ∫[0 to 1] (1/2)e^u du
Again, integrating using the limits of integration:
A_under = [(1/2)e^u] [0 to 1]
A_under = (1/2)e^1 - (1/2)e^0
A_under = (1/2)e - (1/2)
Step 4: Subtract the area below the x-axis from the area above the x-axis to find the total area:
Total Area = A - A_under
Total Area = (1/2)e - (1/2) - [(1/2)e - (1/2)]
Total Area = (1/2)e - (1/2)e + 1/2 - 1/2
Total Area = (1/2)e - (1/2)e + 0
Total Area = 0
Since the total area is 0, none of the given answer choices are correct.