009B Sample Midterm 1, Problem 5

Let 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 f(x)=1-x^2} .

a) Compute the left-hand Riemann sum approximation of $\int _{0}^{3}f(x)~dx$ with $n=3$ boxes.

b) Compute the right-hand Riemann sum approximation of $\int _{0}^{3}f(x)~dx$ with $n=3$ boxes.

c) Express $\int _{0}^{3}f(x)~dx$ as a limit of right-hand Riemann sums (as in the definition of the definite integral). Do not evaluate the limit.

Foundations:
Link to Riemann sums page

Solution:

(a)

Step 1:
Since our interval is $[0,3]$ and we are using 3 rectangles, each rectangle has width 1. So, the left-hand Riemann sum is
$1(f(0)+f(1)+f(2))$.

Step 2:
Thus, the left-hand Riemann sum is
$1(f(0)+f(1)+f(2))=1+0+-3=-2$.

(b)

Step 1:
Since our interval is $[0,3]$ and we are using 3 rectangles, each rectangle has width 1. So, the right-hand Riemann sum is
$1(f(1)+f(2)+f(3))$.

Step 2:
Thus, the right-hand Riemann sum is
$1(f(1)+f(2)+f(3))=0+-3+-8=-11$.

(c)

Step 1:
Let $n$ be the number of rectangles used in the right-hand Riemann sum for $f(x)=1-x^{2}$.
The width of each rectangle is $\Delta x={\frac {3-0}{n}}={\frac {3}{n}}$.

Step 2:
So, the right-hand Riemann sum is
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Delta x{\bigg (}f{\bigg (}1\cdot {\frac {3}{n}}{\bigg )}+f{\bigg (}2\cdot {\frac {3}{n}}{\bigg )}+f{\bigg (}3\cdot {\frac {3}{n}}{\bigg )}+\ldots +f(3){\bigg )}} .
Finally, we let $n$ go to infinity to get a limit.
Thus, $\int _{0}^{3}f(x)~dx$ is equal to 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 \lim_{n\to\infty} \frac{3}{n}\sum_{i=1}^{n}f\bigg(i\frac{3}{n}\bigg)} .

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 -2}
(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 -11}
(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 \lim_{n\to\infty} \frac{3}{n}\sum_{i=1}^{n}f\bigg(i\frac{3}{n}\bigg)}

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