For the function f described by the following table, estimate �ç40 0 f(x)dx using the following. Enter each answer as a sum to show your work.
5 left rectangles
5 right rectangles
5 midpoint rectangles
x 0 4 8 12 16 20 24 28 32 36 40
y 7 -46 -50 45 -7 -26 23 -3 18 -24 19
To estimate ∫₀⁴₀ f(x)dx using rectangles, we can divide the interval [0, 40] into smaller subintervals and approximate the area under the curve using the height of each rectangle multiplied by the width of the subinterval.
Let's calculate the estimates using left rectangles, right rectangles, and midpoint rectangles, each with 5 subintervals.
For left rectangles:
1. Divide the interval [0, 40] into 5 subintervals with equal width.
∆x = (40 - 0) / 5 = 8
2. Calculate the left endpoints of each subinterval.
Left endpoints: {0, 8, 16, 24, 32}
3. Calculate the heights of the rectangles using the given function values.
Heights: {7, -50, -7, 23, 18}
4. Calculate the areas of each rectangle.
Areas: {7 * 8, -50 * 8, -7 * 8, 23 * 8, 18 * 8} = {56, -400, -56, 184, 144}
5. Sum up the areas of the rectangles to get the estimate.
Estimate for left rectangles: 56 + (-400) + (-56) + 184 + 144 = -72
For right rectangles:
1. Divide the interval [0, 40] into 5 subintervals with equal width.
∆x = (40 - 0) / 5 = 8
2. Calculate the right endpoints of each subinterval.
Right endpoints: {8, 16, 24, 32, 40}
3. Calculate the heights of the rectangles using the given function values.
Heights: {-46, -50, 45, -26, 19}
4. Calculate the areas of each rectangle.
Areas: {-46 * 8, -50 * 8, 45 * 8, -26 * 8, 19 * 8} = {-368, -400, 360, -208, 152}
5. Sum up the areas of the rectangles to get the estimate.
Estimate for right rectangles: -368 + (-400) + 360 + (-208) + 152 = -464
For midpoint rectangles:
1. Divide the interval [0, 40] into 5 subintervals with equal width.
∆x = (40 - 0) / 5 = 8
2. Calculate the midpoints of each subinterval.
Midpoints: {4, 12, 20, 28, 36}
3. Calculate the heights of the rectangles using the given function values.
Heights: {-46, -7, -26, -3, -24}
4. Calculate the areas of each rectangle.
Areas: {-46 * 8, -7 * 8, -26 * 8, -3 * 8, -24 * 8} = {-368, -56, -208, -24, -192}
5. Sum up the areas of the rectangles to get the estimate.
Estimate for midpoint rectangles: -368 + (-56) + (-208) + (-24) + (-192) = -848
Therefore, the estimates for ∫₀⁴₀ f(x)dx using 5 left rectangles is -72, using 5 right rectangles is -464, and using 5 midpoint rectangles is -848.