Solving Integral Problems: Step-by-Step Guide for Definite Integrals in Calculus

[Learner123]

Intergral problem! please help!

Homework Statement

$$\oint$$(x: 0 to 1)$$\oint$$(y: $$\sqrt{}(1 - x^2)$$ to e$$\overline{}x$$) xydydx

The region bounded by y = e$$\overline{}x$$, y = $$\sqrt{}(1 - x^2)$$, and x =1
3. The Attempt at a Solution
i got stuck when i came to the part: 1/2 $$\oint$$(x: 0 to 1) (e^(2x) -1 + x^2)xdx
i appreciate any help

 

[slider142]

The symbol you are using is the symbol for a closed line integral. You should be using a normal integral sign: $$\int$$.
Otherwise, since the integral is a linear operator, you have the following sum of integrals:
$$\frac{1}{2}\left(\int xe^{2x} dx - \int x dx + \int x^3 dx\right)$$
Which one is giving you a problem?

 

[Learner123]

the first one xe^(2x) thing
i guess it's intergral by part, but not sure

 

[Learner123]

I tried to do part and this is how i done (for the first intergral):
u = x, du = dx, v = 1/2e^(2x), dv = e^(2x)dx
uv - $$\int$$ vdu
1/2xe^(2x) - $$\int$$ 1/2e^(2x)dx
1/2xe^(2x) - 1/4(e^2 -1 )
x runs from 0 to 1, but 1/2xe^(2x) is not in the intergral part, so how to eliminate x?
very appreciate for more help!

 

[slider142]

1/2xe^(2x) - 1/4(e^2 -1 )

This entire expression is the indefinite integral; the entire expression must be evaluated at the endpoints of the integral if the integral is definite.

 

[Learner123]

got it! i didn't know that after spending 3 calculus classes, what a shame of me! thank you so much for your help and your time.