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Definite Integrals

Fundamental Theorem of Calculus


Evaluate the definite integral \[ \int_{0}^{2}\left( \sin^2 x-x \right)dx - \int_{2}^{0}\left( \cos^2 x-x \right)dx.\]

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

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

Is the above statement true?

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


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