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:
1. The height of each rectangle in the lower Riemann sum is given by choosing the minimum $y$ value of the left and right endpoints of the rectangle.
2. The height of each rectangle in the upper Riemann sum is given by choosing the maximum $y$ value of the left and right endpoints of the rectangle.
3. The area of the region is given by $\int _{a}^{b}y~dx$ for appropriate values $a,b$.

Solution:

(a)

Step 1:
We need to set these two equations equal in order to find the intersection points of these functions.
So, we let $2(-x^{2}+9)=0$. Solving for $x$, we get Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=-3,3} .
This means that we need to calculate the Riemann sums over the interval $[-3,3]$.

Step 2:
Since the length of our interval is $6$ and we are using $3$ rectangles,
each rectangle will have width $2$.
Thus, the lower Riemann sum is
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2(f(-3)+f(-1)+f(3))=2(0+16+0)=32} .

(b)

Step 1:
As in Part (a), the length of our inteval is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6} and
each rectangle will have width Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} . (See Step 1 and 2 for (a))

Step 2:
Thus, the upper Riemann sum is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2(f(-1)+f(-1)+f(1))=2(16+16+16)=96}

(c)

Step 1:
To find the actual area of the region, we need to calculate
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-3}^3 2(-x^2+9)~dx}

Step 2:
We integrate to get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 32}
(b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 96}
(c) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 72}

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