To estimate the integral [tex]\(\int_0^3 x^3 \, dx\)[/tex] using 3 circumscribed rectangles of equal width and to find the absolute value of the resulting error, follow these steps:
1. Determine the exact value of the integral:
The integral [tex]\(\int_0^3 x^3 \, dx\)[/tex] can be calculated analytically. The antiderivative of [tex]\(x^3\)[/tex] is [tex]\(\frac{x^4}{4}\)[/tex]. Therefore,
[tex]\[
\int_0^3 x^3 \, dx = \left. \frac{x^4}{4} \right|_0^3 = \frac{3^4}{4} = \frac{81}{4} = 20.25.
\][/tex]
2. Divide the interval into 3 subintervals:
The interval [tex]\([0, 3]\)[/tex] is divided into 3 subintervals, each of equal width. The width of each rectangle is
[tex]\[
\text{width} = \frac{3 - 0}{3} = 1.
\][/tex]
3. Identify the right endpoints of the subintervals:
Use the right-endpoint method to determine the heights of the rectangles. The right endpoints for each subinterval are [tex]\(1\)[/tex], [tex]\(2\)[/tex], and [tex]\(3\)[/tex].
4. Calculate the height of each rectangle:
The heights of the rectangles are determined by the value of the function at the right endpoints, [tex]\(f(x) = x^3\)[/tex]:
[tex]\[
f(1) = 1^3 = 1,
\][/tex]
[tex]\[
f(2) = 2^3 = 8,
\][/tex]
[tex]\[
f(3) = 3^3 = 27.
\][/tex]
5. Calculate the area of each rectangle:
The area of each rectangle is given by the product of the width and the corresponding height:
[tex]\[
\text{Area of first rectangle} = 1 \cdot 1 = 1,
\][/tex]
[tex]\[
\text{Area of second rectangle} = 1 \cdot 8 = 8,
\][/tex]
[tex]\[
\text{Area of third rectangle} = 1 \cdot 27 = 27.
\][/tex]
6. Sum the areas of the rectangles:
The estimated value of the integral using the sum of the areas of the rectangles is
[tex]\[
\text{Estimated integral} = 1 + 8 + 27 = 36.
\][/tex]
7. Calculate the absolute error:
The absolute error is the difference between the exact value of the integral and the estimated value:
[tex]\[
\text{Error} = \left| 20.25 - 36 \right| = \left| -15.75 \right| = 15.75.
\][/tex]
Therefore, the absolute value of the resulting error is [tex]\(15.75\)[/tex].
Hence, the correct answer is:
[tex]\[
15.75
\][/tex]
