I. Estimating Area Under a Curve Suppose we want to estimate the area under the curve f (x) = x² + 1 between x = 0 and x = 4. TY 16 12 3. Suppose that the area under some curve is divided into n rectangles and we want the area under the curve between a and b. If we divide up the area into n rectangles of equal width, what is the formula for the width, Ax? Ax = = 3 using a right-hand sum. 8 LO 4 .5 1.0 1.5 2.0 2.5 3.0 3.5 To find the shaded area, we will use Riemann rectangles (named for Bernard Riemann). Suppose we draw 4 rectangles to estimate the area under the curve using a left-hand sum. We break up our horizontal distance into 4 equal lengths. We take the total width, 4 - 0 = 4, and divide it by the number of rectangles, 4, to get equal-width rectangles. Thus, each rectangle is width = 1. For the height of each rectangle, we'll use the left- 4-0 4 hand endpoint (the height or y-value of the curve at the left side of the rectangle.) 4. Let's examine a new function. Estimate the area under the curve |f(x) = −4x² +16x + 3 between x = 0 and x a. If we use 6 rectangles, what is the width Ax? b. The height of each rectangle is given by the right-hand endpoint because this is a right-hand sum. Draw the 6 Riemann rectangles for the right-hand sum. SHADE INSIDE EACH RECTANGLE. Ty 16 12 .5 1.0 1.5 2.0 2.5

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Find x. All dimensions are in millimeters.

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