(a) Estimate the area under the graph of f(x)=cosx from x=0 to x=π/ 2 using four approximating r

step 1 Okay, so for the part a, the area under the curve is equal to pi over 8 times cos of pi over 8 p

(a) Estimate the area under the graph of f(x)=cosx from x=0...

(a) Estimate the area under the graph of $f(x)=\cos x$ from $x=0$ to $x=\pi / 2$ using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate?
(b) Repeat part (a) using left endpoints.

Key Concepts

Left and Right Endpoint Approximations

These are specific methods for determining the height of each rectangle in a Riemann sum. Left endpoint approximations use the function value at the beginning of each subinterval, while right endpoint approximations use the value at the end. The choice of endpoint affects the accuracy of the approximation, especially when the function is increasing or decreasing.

Overestimation and Underestimation

This concept deals with how well the Riemann sum approximates the actual area under a curve. Depending on whether the function is increasing or decreasing over the interval, using left or right endpoints can systematically yield an approximation that is too high (overestimate) or too low (underestimate) compared to the true value of the definite integral.

Definite Integral

A definite integral represents the area under the curve of a continuous function over a specific interval. It’s a fundamental concept in calculus used to calculate accumulation of quantities, and its precise value is obtained as the limit of Riemann sums as the subintervals' widths approach zero.

Riemann Sums

Riemann sums approximate the area under a curve by partitioning the interval into smaller subintervals, constructing rectangles over each subinterval, and then summing their areas. The approximation improves with a finer partition, i.e., more and smaller subintervals.

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