009B Sample Final 1, Problem 1

Consider the region bounded by the following two functions:

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 y=2(-x^2+9)} and 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 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
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)
(b)
(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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