Approximate the area under the curve from using approximating rectangles of equal widths and right endpoints. The choices are rounded to the nearest hundredth.

Question

Approximate the area under the curve  from  using  approximating rectangles of equal widths and right endpoints. The choices are rounded to the nearest hundredth.

Quizplus · Accepted Answer

Using 6 rectangles of equal width, we have:

$$\Delta x = \frac{6.4 - 1}{6} \approx 1.07$$

The right endpoints of the rectangles are: $$1.07, 2.14, 3.21, 4.28, 5.35, 6.42$$

The area of each rectangle is:

$$\Delta A_i \approx f(x_i) \Delta x$$

where $x_i$ is the right endpoint of the $i$th rectangle.

Thus, we have:

$$\begin{aligned} \Delta A_1 &\approx f(1.07) 	imes 1.07 \approx 11.06 \ \Delta A_2 &\approx f(2.14) 	imes 1.07 \approx 25.94 \ \Delta A_3 &\approx f(3.21) 	imes 1.07 \approx 51.34 \ \Delta A_4 &\approx f(4.28) 	imes 1.07 \approx 87.35 \ \Delta A_5 &\approx f(5.35) 	imes 1.07 \approx 132.97 \ \Delta A_6 &\approx f(6.42) 	imes 1.07 \approx 187.20 \end{aligned}$$

Summing up the areas of the rectangles gives an approximation of the integral:

$$\begin{aligned} A &\approx \Delta A_1 + \Delta A_2 + \Delta A_3 + \Delta A_4 + \Delta A_5 + \Delta A_6 \ &\approx 11.06 + 25.94 + 51.34 + 87.35 + 132.97 + 187.20 \ &\approx 495.86 \end{aligned}$$

Therefore, the closest choice is C) 495.60.

Approximate the Area Under the Curve from Using