1. ## Area of shaded region

I posted a question similar to this a minute ago, but I wanted to double check and make sure I'm doing it right. The question is find the shaded region of $\displaystyle 7x(x^2-64)$ There is a picture, but I can't display it here. The bounds are 0 to 8 and the second equation is y = 0. So, you take the top curve (0) - the bottom curve $\displaystyle 7x(x^2-64)$. Leaving you with $\displaystyle -7x^3-448x$. From there you integrate which gives $\displaystyle -7/4x^4 - 224x^2$. This is where I get confused with definite integration. Do I put in 8 for the first x and then 0 for the second x? E.g. $\displaystyle -7/4(8)^4 - 224(0)^2$ Or do I do two whole separate equations E.g. $\displaystyle (-7/4(8)^4 - 224(8)^2) - (-7/4(0)^4 - 224(0)^2)$ . Can someone explain? Thanks

2. Originally Posted by zsig013
I posted a question similar to this a minute ago, but I wanted to double check and make sure I'm doing it right. The question is find the shaded region of $\displaystyle 7x(x^2-64)$ There is a picture, but I can't display it here. The bounds are 0 to 8 and the second equation is y = 0. So, you take the top curve (0) - the bottom curve $\displaystyle 7x(x^2-64)$. Leaving you with $\displaystyle -7x^3-448x$. From there you integrate which gives $\displaystyle -7/4x^4 - 224x^2$. This is where I get confused with definite integration. Do I put in 8 for the first x and then 0 for the second x? E.g. $\displaystyle -7/4(8)^4 - 224(0)^2$ Or do I do two whole separate equations E.g. $\displaystyle (-7/4(8)^4 - 224(8)^2) - (-7/4(0)^4 - 224(0)^2)$ . Can someone explain? Thanks
Remember to evaluate the definite integral, we need to use the Fundamental Theorem of Calculus:
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
where $\displaystyle F(x)$ is the antiderivative of $\displaystyle f(x)$.

In your case, we have the antiderivative $\displaystyle -\frac{7}{4}x^4-224x^2$. Then we need to find $\displaystyle F(8)-F(0)=(-\frac{7}{4}(8)^4-224(8)^2)-(-\frac{7}{4}(0)^4-224(0)^2)$.

I hope this clarified things!!!