Definite Integrals

Fundamental Theorem of Calculus


Evaluate the definite integral 02(sin2xx)dx20(cos2xx)dx. \int_{0}^{2}\left( \sin^2 x-x \right)dx - \int_{2}^{0}\left( \cos^2 x-x \right)dx.

If f(x)=xx+1(2t2+t) dt\displaystyle f(x) = \int_{x}^{x+1} (2t^2+t)\ dt, what is the value of f(7)f'(7)?

The volume of water in a container at a height of x cmx\ \mbox{cm} from the bottom, can be expressed as V(x)=x33x2+4x V(x) = x^3-3x^2+4x in cm3\mbox{cm}^3. If the surface area of water at heights of y cmy \ \mbox{cm} and 16y cm\frac{1}{6}y \ \mbox{cm} are the same, then y=aby=\frac{a}{b}, where aa and bb are coprime positive integers. What is the value of a+ba+b?

Is the above statement true?

Given function g(x)=0x1+t3dt,g(x)=\int_{0}^{x} \sqrt{1+t^{3}}dt, what is g(x)?g'(x)?


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