Use a Riemann sum with 4 rectangles of equal width to approximate the area between y equals x cubed plus 2 and the x-axis on the interval left square bracket short dash 1 comma 1 right square bracket. Use the right-hand endpoint of each subinterval.A. 9 space u n i t s squaredB. 3.5 space u n i t s squaredC. 4.5 space u n i t s squaredD. 8.0625 space u n i t s squared
Question
Use a Riemann sum with 4 rectangles of equal width to approximate the area between y equals x cubed plus 2 and the x-axis on the interval left square bracket short dash 1 comma 1 right square bracket. Use the right-hand endpoint of each subinterval.A. 9 space u n i t s squaredB. 3.5 space u n i t s squaredC. 4.5 space u n i t s squaredD. 8.0625 space u n i t s squared
Solution
The problem is asking us to approximate the area under the curve y = x^3 + 2 from x = -1 to x = 1 using a Riemann sum with 4 rectangles. We will use the right-hand endpoint of each subinterval.
Step 1: Determine the width of each rectangle. The total width of the interval is 1 - (-1) = 2. Since we are using 4 rectangles, each rectangle will have a width of 2/4 = 0.5.
Step 2: Determine the right-hand endpoints of each subinterval. The right-hand endpoints of the subintervals are -0.5, 0, 0.5, and 1.
Step 3: Evaluate the function at each right-hand endpoint. The function values at the right-hand endpoints are (-0.5)^3 + 2 = 1.875, (0)^3 + 2 = 2, (0.5)^3 + 2 = 2.125, and (1)^3 + 2 = 3.
Step 4: Multiply each function value by the width of the rectangles and add up the results. The Riemann sum is (1.875)(0.5) + (2)(0.5) + (2.125)(0.5) + (3)(0.5) = 0.9375 + 1 + 1.0625 + 1.5 = 4.5.
So, the Riemann sum approximation for the area under the curve is 4.5 units squared. The correct answer is C. 4.5 space units squared.
Similar Questions
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