Estimate the area under the graph of f ( x ) = 3 cos( x ) from x = 0 to x = ? /2

Question

Estimate the area under the graph of f(x) = 3 cos(x) from x = 0 to x = ?/2 using four approximating rectangles and right endpoints.

Estimate the area under the graph of f(x) = 3 cos(x) from x = 0 to x = ?/2 using four approximating rectangles and LEFT endpoints.

We wish to find R4 = [f(x1) + f(x2) + f(x3) + f(x4)](pi/8). Since x1, x2, x3, X4 represent the right-hand endpoints of the four sub-intervals of [0, pi/2], then we must have the following. We wish to find L4 = [f(x0) +f(X1)+ f(x2) + f(x3) + f(x4)](pi/8). Since x0, x1, x2, x3, represent the right-hand endpoints of the four sub-intervals of [0, pi/2], then we must have the following.

Explanation / Answer

You want to split [0, pi/2] into 4 rectangles. The space between endpoints (delta-x) is given as pi/8 so you add pi/8 in between endpoints. Right endpoints (x1 to x4) are given as:

x0 = 0

x1 = 0 + pi/8 = pi/8

x2 = pi/8 + pi/8 = 2pi/8 = pi/4

x3 = 2pi/8 + pi/8 = 3pi/8

x4 = 3pi/8 + pi/8 = 4pi/8 = pi/2

Left endpoints (x0 to x3) are given as:

x0 = 0

x1 = pi/8

x2 = pi/8 + pi/8 = 2pi/8 = pi/4

x3 = 2pi/8 + pi/8 = 3pi/8

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