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Definite Integrals

The definite integral of a function computes the area under the graph of its curve, allowing us to calculate areas and volumes that are not easily done using geometry alone.

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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