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1. Find the area bounded above by 𝑦 = √𝑥 and below by the x-axis and 𝑦 = 𝑥 − 2 in the interval [0,
4].
i. First plot all the functions.
ii. Find the area by computing the relevant finite integrals with respect to x
iii. Repeat ii by integrating with respect to y.
2. Find the volume of the solid generated by the revolving region between the parabola x = y2+1 and
the line x = 3 about the line x = 3.
Below figure shows the region, a typical radius, and the generated solid.
Definition 1: Area between two curves
If f and g are continuous with f(x) ≥ g(x) throughout [a, b], then the area of the region between the
curves f(x) and g(x) from a to b is the integral of (f – g) from a to b.
Definition 2: Volume by Disks for rotation about the x-axis
The solid generated by rotating (or revolving) a plane region about an axis in its plane is called a solid
of revolution. The volume of such solid can be found by finding the cross sectional area A(x), that is
the area of the disk of radius R(x), the distance of the planar regions’ boundary from the x-axis of the
revolution. V (volume) = ∫ 𝐴(𝑥)𝑑𝑥 = ∫ 𝜋 [𝑅(𝑥)]
Relevant MATLAB function: integral(fun,xmin,xmax)
1. Find the area bounded above by 𝑦 = √𝑥 and below by the x-axis and 𝑦 = 𝑥 − 2 in the interval [0,
4].
i. First plot all the functions.
ii. Find the area by computing the relevant finite integrals with respect to x
iii. Repeat ii by integrating with respect to y.
2. Find the volume of the solid generated by the revolving region between the parabola x = y2+1 and
the line x = 3 about the line x = 3.
Below figure shows the region, a typical radius, and the generated solid.
Definition 1: Area between two curves
If f and g are continuous with f(x) ≥ g(x) throughout [a, b], then the area of the region between the
curves f(x) and g(x) from a to b is the integral of (f – g) from a to b.
Definition 2: Volume by Disks for rotation about the x-axis
The solid generated by rotating (or revolving) a plane region about an axis in its plane is called a solid
of revolution. The volume of such solid can be found by finding the cross sectional area A(x), that is
the area of the disk of radius R(x), the distance of the planar regions’ boundary from the x-axis of the
revolution. V (volume) = ∫ 𝐴(𝑥)𝑑𝑥 = ∫ 𝜋 [𝑅(𝑥)]
Relevant MATLAB function: integral(fun,xmin,xmax)