Evaluate the definite integral by the limit definition. 6x2 dx Given f(x) dx = 1

Question

Evaluate the definite integral by the limit definition. 6x2 dx Given f(x) dx = 10 and g(x) dx =-4, evaluate the following. Sketch the region whose area is given by the definite integral. (9-|x|) dx Use a geometric formula to evaluate the integral. Set up a definite integral that yields the area of the region. (Do not evaluate the integral.) f(x) = cos(x) The graph of f consists of line segments and a semicircle, as shown in the figure. Evaluate each definite integral by using geometric formulas.

Explanation / Answer

3) the shaded region is the bottom left one.

the integral is the triangles area = 1/2 (18) (9) = 81 unit^2

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4) integral (from 0 to pi/2) cosx dx

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5)

a) = - (pi r^2 / 4) = - pi (2^2)/4 = - pi

b) 1/2 (6 - 2) x 2 = 4

c) -1 x [1/2 (1) (2) + pi (2^2) /2 ] = -1 - 2pi

d) its (c) + (d) = -1 - 2pi + 4 = 3 - 2pi

e) | -1 - 2pi | + 4 = 1 + 2pi + 4 = 5 + 2pi

f) it's 3 - 2pi + 2(6 - -4) = 23 - 2pi

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