# Thread: Integral Question using Fundamental Theorem of Calculus

1. ## Integral Question using Fundamental Theorem of Calculus

Hi everyone,

I am working on this question for calculus, but I am a bit confused by the y's in the range of the integral.

Here is the question, which asks you to find p'(y).

I am a bit confused as to what to do next, would subbing 9y for x be all I need to do for this question? (Fundamental theorem of calculus part 1)

Thanks a lot.

2. Just plug the "y's" in like they were numbers and your answer will be a function. Here's an example:

$\displaystyle \int_{2y}^{3y}2xdx=(x^2)|_{2y}^{3y}=(3y)^2-(2y)^2$

3. Hi again,

VonNemo19, wouldn't that be the method for finding the area (definite integral)? I think the question was asking for the derivative of the original function.

4. No, the question is asking you to take the integral with respect to x. So, pretend like you don't see the limits of integration (the y's) and find the integral. Then substitute the y's in.

BTW the fundamental theorem states that $\displaystyle \int_a^bf'(x)dx=f(b)-f(a)$.

Another example...$\displaystyle \int_x^{x^2}(y+y^2)dy=\left(\frac{y^2}{2}+\frac{y^ 3}{3}\right)\Big|_x^{x^2}=\left[\frac{(X^2)^2}{2}+\frac{(X^2)^3}{3}\right]-\left[\frac{(X)^2}{2}+\frac{(X)^3}{3}\right]$

5. Oooppps. Did you just now put that apostrophe in there? p' is very different than p.

p' is simply...the integrand
I coulda swore there was no apostrophe there earlier

6. Example.

$\displaystyle f(x)=\int_y^22xdx=x^2\Big|_y^2=2^2-y^2$

Now, $\displaystyle f'(y)=-2y$

7. Oh, okay thanks!

8. Here is a simple example to show how to do it.

If $\displaystyle p(y) =\displaystyle \int_{8y}^{9y} {(x^2 + 3x - 1)dx}$ then $\displaystyle p'(y) = \left[ {9\left( {(9y)^2 + 3\left( {9y} \right) - 1} \right)} \right] - \left[ {8\left( {(8y)^2 + 3\left( {8y} \right) - 1} \right)} \right]$

Here it idea: if $\displaystyle p(y) =\displaystyle \int_{f(y)}^{g(y)} {h(x)dx}$ then $\displaystyle p'(y) = \left[ {h(g(x))g'(x)} \right] - \left[ {h(f(x))f'(x)} \right]$

9. Dude, I'm really sorry. I messed up like 6 times on this problem. I need to learn how to read better.