Estimate the area from 0 to 5 under the graph of $f ( x ) = 49 - x ^ { 2 }$ using five approximating rectangles and right endpoints.
A) 191
B) 188
C) 190
D) 189
E) 192

Question

Quizplus · Accepted Answer

Using five approximating rectangles with right endpoints, the width of each rectangle is 1 since the interval from 0 to 5 is divided into five equal parts. The height of each rectangle can be found by evaluating the function at the right endpoint of each interval.

The height of the first rectangle is $f(1)=49-1^2=48$. 
The height of the second rectangle is $f(2)=49-2^2=45$. 
The height of the third rectangle is $f(3)=49-3^2=40$. 
The height of the fourth rectangle is $f(4)=49-4^2=33$. 
The height of the fifth rectangle is $f(5)=49-5^2=24$.

The area of each rectangle is therefore:

\begin{align*}
1\cdot 48 &= 48\
1\cdot 45 &= 45\
1\cdot 40 &= 40\
1\cdot 33 &= 33\
1\cdot 24 &= 24\
\end{align*}

The total area is the sum of the areas of the rectangles:
$$48+45+40+33+24=190$$

Therefore, the best choice is C) 190.

Estimate the Area from 0 to 5 Under the Graph f(x)=49−x2f ( x ) = 49 - x ^ { 2 }f(x)=49−x2

Estimate the Area from 0 to 5 Under the Graph $f ( x ) = 49 - x ^ { 2 }$