To estimate the value of the definite integral [tex]\(\int_1^9 4x^3 \, dx\)[/tex] using a Riemann Sum with an overestimation for [tex]\( n = 4 \)[/tex] subdivisions, follow these steps:
1. Determine the width of each subinterval (Δx):
[tex]\[
\Delta x = \frac{b - a}{n} = \frac{9 - 1}{4} = \frac{8}{4} = 2
\][/tex]
2. Define the function to be integrated [tex]\( f(x) = 4x^3 \)[/tex].
3. Determine the sample points, specifically the right endpoints for the overestimate:
- The right endpoints are calculated as follows [tex]\( x_i = a + i\Delta x \)[/tex] for [tex]\( i = 1, 2, 3, 4 \)[/tex].
- Calculating the right endpoints:
[tex]\[
x_1 = 1 + 1 \cdot 2 = 3
\][/tex]
[tex]\[
x_2 = 1 + 2 \cdot 2 = 5
\][/tex]
[tex]\[
x_3 = 1 + 3 \cdot 2 = 7
\][/tex]
[tex]\[
x_4 = 1 + 4 \cdot 2 = 9
\][/tex]
- Therefore, our right endpoints are [tex]\(3, 5, 7,\)[/tex] and [tex]\(9\)[/tex].
4. Evaluate the function at these right endpoints:
[tex]\[
f(3) = 4 \cdot 3^3 = 4 \cdot 27 = 108
\][/tex]
[tex]\[
f(5) = 4 \cdot 5^3 = 4 \cdot 125 = 500
\][/tex]
[tex]\[
f(7) = 4 \cdot 7^3 = 4 \cdot 343 = 1372
\][/tex]
[tex]\[
f(9) = 4 \cdot 9^3 = 4 \cdot 729 = 2916
\][/tex]
5. Calculate the Riemann sum (overestimate):
- Multiply each function value at the right endpoint by the width of the subinterval [tex]\( \Delta x \)[/tex]:
[tex]\[
\text{Riemann Sum} = \Delta x \left[f(3) + f(5) + f(7) + f(9)\right]
\][/tex]
- Substitute the values:
[tex]\[
\text{Riemann Sum} = 2 \left[108 + 500 + 1372 + 2916\right]
\][/tex]
- Now sum the values inside the brackets:
[tex]\[
108 + 500 + 1372 + 2916 = 4896
\][/tex]
- Multiply by [tex]\( \Delta x \)[/tex]:
[tex]\[
2 \times 4896 = 9792
\][/tex]
Therefore, the estimate of the integral using a Riemann Sum with [tex]\( n = 4 \)[/tex] subdivisions, which is an overestimate, is:
[tex]\[
\boxed{9792}
\][/tex]
