Fundemental Theorm of Calculus : Integrals and Areas (8 Problems)
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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.
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Solution Summary
Eight integrals and areas are found and the details are discussed in the solution.
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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
= ...
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