009B Sample Final 1, Problem 1

Consider the region bounded by the following two functions:

$y=2(-x^{2}+9)$ and $y=0$

a) Using the lower sum with three rectangles having equal width , approximate the area.

b) Using the upper sum with three rectangles having equal width, approximate the area.

c) Find the actual area of the region.

Foundations:
Link to Riemann sums page

Solution:

(a)

Step 1:

Step 2:

(b)

Step 1:

Step 2:

Step 3:

(c)

Step 1:
To find the actual area of the region, we need to calculate
$\int _{-3}^{3}2(-x^{2}+9)~dx$

Step 2:
We integrate to get
${\begin{array}{rcl}\displaystyle {\int _{-3}^{3}2(-x^{2}+9)~dx}&=&\displaystyle {2{\bigg (}{\frac {-x^{3}}{3}}+9x{\bigg )}{\bigg |}_{-3}^{3}}\\&&\\&=&\displaystyle {2{\bigg (}{\frac {-3^{3}}{3}}+9\times 3{\bigg )}-2{\bigg (}{\frac {-(-3)^{3}}{3}}+9(-3){\bigg )}}\\&&\\&=&\displaystyle {2(-9+27)-2(9-27)}\\&&\\&=&\displaystyle {2(18)-2(-18)}\\&&\\&=&\displaystyle {72}\\\end{array}}$

Final Answer:
(a)
(b)
(c) $72$

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