Question

Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown in the figure below. Evaluate the integral exactly. Use your work to answer the questions below.
http://www.webassign.net/userimages/8.1.007.jpg?db=v4net&id=53285

What is the approximate area of the strip with respect to y?

Answer 1

what language are we speaking here? MATH lol?

Answer 2

I need help. not jokes

Answer 3

The shaded region would be partitioned in the following way:
\[P=\left\{y_0,y_1,\ldots,y_{k-1},y_k\right\},\]
where y_0 and y_k are the points of intersection of the two curves.

\[\text{Denote $x=y$ and $x=y^2$ by $f(y)$ and $g(y)$, respectively.}\\\text{So, $f(y)=y$ and $g(y)=y^2$.}\]
As a Riemann sum:
\[\large A\approx\sum_{k=1}^{n}\left(f(y^*_k)-g(y^*_k)\right)\Delta y,\\
\text{where }\Delta y=y_{i}-y_{i-1}, \;\;\;\;\;(*)\\
\text{and }y^*_k\in\left[y_{i-1},y_{i}\right]\]