Using n = 4 equal-width rectangles, approximate the integral

Question

Using n = 4 equal-width rectangles, approximate  . Use the left end-point of each sub-interval to determine the height of each rectangle.

Answer 1

The rectangle bases are each 1; the left sides are -2, -1, 0, 1 and f(x) is the height.

Here are the areas:

x b f(x) area

-2 1 12 12

-1 1 9 9

0 1 8 8

1 1 9 9

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TOTAL: 38

Integral is approximately 38.

Answer 2

With 4 equal-width rectangles across an interval of width 4, each one has width 1. So in the rectangle area formula a = b*h, the base b is 1 in each case.

For the height, evaluate the integrand at x = -2, -1, 0 and 1:

Leftmost rectangle's area = 1*(2²+8) = 12

Area of next rectangle = 1*[(-1)²+8] = 10

Area of next rectangle = 1*(0²+8) = 8

Area of last rectangle = 1*(1²+8) = 10

Our approximation of the integral is 40 (the sum of the areas above).