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Fundemental Theorm of Calculus : Integrals and Areas (8 Problems)

Find the area of the region

1.) y=x- x^2 points (0,1/4) (1,0)

2.) y=1/x^2 points ((1,1) (2, 1/2)

Find and evaluate the integral

1.) integral 1 to 0 2xdx

2.) integral 0 to -1 (2x +1)dx

3.) integral 1 to -1 (2t -1)^2 dt

4.) integral 5 to 2 (-3x +4) dx

5.) integral 4 to 0 1/sq rt 2x +1 dx

6.) integral 2 to 0 x/ sq rt 1+2x^2 dx

Solution Preview

1.) y = x - x^2

points (o,1/4) (1,0)

Area under the curve y=f(x) and the points x=a and x=b is given by

A = Int ydx with the limit from a to b

Here, A = Int (x - x^2 ) from 0 to 1

A = x^2/2 - x^3/3 Applying limits, ==> 1^2/2 - 1^3/3 - 0 = 1/2 - 1/3 = 1/6 --ans

2.) y=1/x^2

points ((1,1) (2, 1/2)

similarly, A = Int(1/x^2) limit from x = 1 to x=2

= -1/x applying limits

= ...

Solution Summary

Eight integrals and areas are found.

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