Approximate the area of the region under the function below on the interval $ [0,2] $ using 8 rectangles. Round your answer to two decimals.
$f(x)=\frac{1}{3} x^{4}$
A) $ 5.71 $ square units
B) $ 2.86 $ square units
C) $ 3.69 $ square units
D) $ 8.57 $ square units
E) $ 17.13 $ square units

Question

Quizplus · Accepted Answer

Using the left endpoint method with 8 rectangles, we have a width of $0.25$ and the heights of the rectangles are $f(0), f(0.25), f(0.5), \ldots, f(1.75), f(2)$. Evaluating these heights, we get: \begin{align*}
f(0) &= 0 \
f(0.25) &= \frac{1}{3}(0.25)^4 = 0.00026 \
f(0.5) &= \frac{1}{3}(0.5)^4 = 0.0625 \
f(0.75) &= \frac{1}{3}(0.75)^4 \approx 0.352 \
f(1) &= \frac{1}{3}(1)^4 = \frac{1}{3} \
f(1.25) &= \frac{1}{3}(1.25)^4 \approx 1.952 \
f(1.5) &= \frac{1}{3}(1.5)^4 \approx 4.196 \
f(1.75) &= \frac{1}{3}(1.75)^4 \approx 7.015 \
f(2) &= \frac{1}{3}(2)^4 = \frac{16}{3}
\end{align*} Summing the areas of the rectangles, we get $$0.25\left(f(0)+f(0.25)+f(0.5)+f(0.75)+f(1)+f(1.25)+f(1.5)+f(1.75)\right)$$ $$=0.25(0+0.00026+0.0625+0.352+\frac{1}{3}+1.952+4.196+7.015)$$ $$\approx\boxed{2.86}.$$

Approximate the Area of the Region Under the Function Below [0,2] [0,2] [0,2]

Approximate the Area of the Region Under the Function Below $[0,2]$