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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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