1. Intergration Help!

Question:
Find the area enclosed between the curves $\displaystyle y = x^2 + 7$ and $\displaystyle y = 2x^2 + 3$.

Attempt:

$\displaystyle [ y = x^2 + 7 ] \times 2$
$\displaystyle [ y = 2x^2 + 3 ]$

$\displaystyle 2y = 2x^2 + 14$
$\displaystyle y = 2x^2 + 3$
$\displaystyle y = 11$

$\displaystyle 11 = 2x^2 + 3$
$\displaystyle 11 - 3 = 2x^2$
$\displaystyle 8 = 2x^2$
$\displaystyle \frac{8}{2} = x^2$
$\displaystyle 4 = x^2$
$\displaystyle x = \sqrt{4}$
$\displaystyle x = 2$

$\displaystyle y = 11 , x = 2$

$\displaystyle y = x^2 +7$

$\displaystyle = \frac{x^{2+1}}{2+1} +7x$

$\displaystyle = \frac{x^3}{3} + 7x$

$\displaystyle = (\frac{1}{3}x^3 + 7x)^2_0$

$\displaystyle = (\frac{1}{3}\times2^3 + 7\times2) - (\frac{1}{3}\times0^3 + 7\times0)$

$\displaystyle = 16\frac{2}{3} - 0$

$\displaystyle = 16\frac{2}{3}$

$\displaystyle y = 2x^2 + 3$

$\displaystyle = \frac{2x^{2+1}}{2+1} + 3x$

$\displaystyle = \frac{2x^3}{3} + 3x$

$\displaystyle = (\frac{2}{3}x^3 + 3x)^2_0$

$\displaystyle = (\frac{2}{3}\times2^3 + 3\times2) - (\frac{2}{3}\times0^3 + 3\times0)$

$\displaystyle = 11\frac{1}{3}$

Enclosed Area = $\displaystyle 16\frac{2}{3} - 11\frac{1}{3}$
$\displaystyle = 5\frac{1}{3}$

Answer in the text book is $\displaystyle 10\frac{2}{3}$

2. First, find the points where the two functions intersect, which is to say you have to define the "ends" of the enclosed area. Both functions are equal to y, so set them equal to one another:

$\displaystyle x^2 + 7 = 2x^2 + 3$

Solving that gives $\displaystyle x= \pm 2$ - remember $\displaystyle \sqrt{4} = \pm 2$ - I think this is the step you missed.

In the interval (-2,2), $\displaystyle x^2 + 7$ has a greater value than $\displaystyle 2x^2 + 3$. In other words, $\displaystyle x^2 + 7$ will be the "top" function.

You're going to have to integrate with respect to x as x goes from -2 to 2, as those are the "ends" of the region of intersection. To find the area enclosed between two curves, you integrate $\displaystyle f(x) - g(x)$ where $\displaystyle f(x)$ is the "top" function and $\displaystyle g(x)$ is the "bottom" function.

$\displaystyle \int^2_{-2} [(x^2 + 7) - (2x^2 + 3)] dx$

$\displaystyle = \int_{-2}^2 (-x^2 + 4) dx$

$\displaystyle = - \frac{1}{3}x^3 + 4x]^2_{-2}$

$\displaystyle = (- \frac{8}{3} + 8) - ( \frac{8}{3} - 8)$

$\displaystyle = 10 \frac{2}{3}$.

Hope this helps!