To approximate the area under the curve of the function [tex]\( f(x) = 2x^2 - 1 \)[/tex] between [tex]\( a = 0 \)[/tex] and [tex]\( b = 8 \)[/tex] using a right-hand sum with 4 intervals, we can follow these steps:
1. Determine the width of each interval [tex]\( \Delta x \)[/tex]:
[tex]\[
\Delta x = \frac{b - a}{n} = \frac{8 - 0}{4} = 2
\][/tex]
2. Identify the right-hand endpoints of each interval:
Since we are using a right-hand sum, the x-values we'll use for our calculations are the endpoints on the right side of each interval:
[tex]\[
x_1 = a + \Delta x \times 1 = 0 + 2 \times 1 = 2
\][/tex]
[tex]\[
x_2 = a + \Delta x \times 2 = 0 + 2 \times 2 = 4
\][/tex]
[tex]\[
x_3 = a + \Delta x \times 3 = 0 + 2 \times 3 = 6
\][/tex]
[tex]\[
x_4 = a + \Delta x \times 4 = 0 + 2 \times 4 = 8
\][/tex]
3. Calculate the function value at each right-hand endpoint:
[tex]\[
f(x_1) = f(2) = 2 \times 2^2 - 1 = 2 \times 4 - 1 = 8 - 1 = 7
\][/tex]
[tex]\[
f(x_2) = f(4) = 2 \times 4^2 - 1 = 2 \times 16 - 1 = 32 - 1 = 31
\][/tex]
[tex]\[
f(x_3) = f(6) = 2 \times 6^2 - 1 = 2 \times 36 - 1 = 72 - 1 = 71
\][/tex]
[tex]\[
f(x_4) = f(8) = 2 \times 8^2 - 1 = 2 \times 64 - 1 = 128 - 1 = 127
\][/tex]
4. Multiply each function value by the width of the interval [tex]\( \Delta x \)[/tex]:
[tex]\[
f(x_1) \times \Delta x = 7 \times 2 = 14
\][/tex]
[tex]\[
f(x_2) \times \Delta x = 31 \times 2 = 62
\][/tex]
[tex]\[
f(x_3) \times \Delta x = 71 \times 2 = 142
\][/tex]
[tex]\[
f(x_4) \times \Delta x = 127 \times 2 = 254
\][/tex]
5. Sum up these products to get the total approximate area:
[tex]\[
\text{Area} \approx 14 + 62 + 142 + 254 = 472
\][/tex]
Therefore, the approximate area under the curve between [tex]\( x = 0 \)[/tex] and [tex]\( x = 8 \)[/tex] using a right-hand sum with 4 intervals is [tex]\( 472 \)[/tex].
