Application of Integrals: Problems with Solutions

By Prof. Hernando Guzman Jaimes (University of Zulia - Maracaibo, Venezuela)
Problem 1
Write the definite integral that gives the area of the region between $y_{1}, y_{2}$
$y_{1}=x^{2}+2x+1, y_{2}=2x+5$.
What is the area?


Problem 2
Find the integral that allows to determine the area of the region between
$y_{1}=x^{2}-4x+3$ and $y_{2}=-x^{2}+2x+3$
What is the area?



Problem 3
Consider the graph, calculate the area of the given region.
$y_{1}=3(x^{3}-x), y_{2}=0$


Problem 4
Find the area of the figure between the functions
$f(x)=-x^{2}+\frac{9}{2}x+1$ and $g(x)=\frac{1}{2}x+1$


Problem 5
Determine the area of the region limited by $y=x, y=2-x, y=0$.


Problem 6
Given functions $f(x)=\sqrt{x}+3,~g(x)=\frac{1}{2}x+3$.
Determine the area of the region obtained between them.


Problem 7
Calculate the area enclosed by the following functions
$f(x)=\sqrt[3]{x-1}$ and $g(x)=x-1$.


Problem 8
Find the area of the region determined by the following graphs
$f(x)=x^{2}-4x+3$ and $g(x)=3+4x-x^{2}$.


Problem 9
Determine the area of the region bounded by $y=x^{4}-2x^{2}$ and $y=2x^{2}$


Problem 10
Determine the area of the region bounded by $f(x)=x^{4}-4x^{2}$ and $g(x)=x^{2}-4$.



Problem 11
Find the volume of the solid formed by revolving the region bounded by the curve
$y=4-x^{2}$ about the x−axis.


Problem 12
Find the volume of the solid formed by revolving the region bounded by the curve $y=\sqrt{x}$, $x \in [0, 3]$ about the x−axis.


Problem 13
Find the volume of the solid formed by revolving the region bounded by the curve
$f(x)=\sqrt{\sin x}$, $0< x < \pi $ about the x-axis.
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