Applications of Integration
Exercise 1 : Area Bounded by Parabola
Calculate the area bounded by the positive branch of the parabola $y^2 = 25x$, the $x$-axis and the ordinates where $x = 0$ and $x = 36$.
Exercise 2 : Area Between a Symmetric Curve and the x-axis
Calculate the area bounded by the positive branch of the curve $y^2 = (7 - x)(5 + x)$, the $x$-axis and the ordinates where $x = -5$ and $x = 1$.
Exercise 3 : Area Under a Parabola between Roots
Calculate the area bounded by the parabola $20y = 3(2x^2 - 3x - 5)$ and the $x$-axis between the points where the curve cuts the $x$-axis.
Exercise 4 : Area under a Reciprocal Quadratic Curve
Calculate the area bounded by the curve $y^2(x^2 + 6x - 55) = 1$, the $x$-axis and the ordinates where $x = 7$ and $x = 14$.
Exercise 5 : Sketch and Area of a Cubic Polynomial
Sketch the curve $y = 2x^3 - 15x^2 + 24x + 25$ between $x = 0$ and $x = 4$ and then calculate the area enclosed by the ordinates at these points, the $x$-axis and the portion of the curve.
Exercise 6 : Polar Area for a Rectangular Hyperbola
Calculate the area bounded by the hyperbola $r^2 \cos 2\theta = 9$ and the radial lines $\theta = 0$ and $\theta = 30^\circ$.
Exercise 7 : Area of a Rose Curve
Calculate the entire area of the curve $r = 3.5 \sin 2\theta$.
Exercise 8 : Intersection Areas
Calculate the area bounded by the following curves:
- $y^2 = 4x$ and $x^2 = 6y$
- $y = 4 - x^2$, $y = 4 - 4x$ and the line joining the points $(-2, -6)$ and $(4, 6)$
- $y = 6 + 4x - x^2$ and the line joining the points $(-2, -6)$ and $(4, 6)$
Exercise 9 : Surface Area of Revolution — Cubic
Calculate the area of the surface generated by the revolution of the curve $y = x^3$ about the $x$-axis between the ordinates $x = 0.5$ and $x = 0$.
Exercise 10 : Surface Area and Volume from a Shifted Parabola
The curve $y = x(6 - x) - 7.56$ is rotated about the $x$-axis between the points where it crosses the $x$-axis. Calculate the surface area and the volume of the solid thus generated.
Exercise 11 : Surface Area and Volume of a Cycloid of Revolution
Calculate the surface area and the volume generated by rotating the cycloid $x = \theta - \sin \theta$, $y = 1 - \cos \theta$ about the $x$-axis.
Exercise 12 : Volume from Revolving an Ellipse (Ellipsoid)
Calculate the volume generated by revolving the ellipse $x^2/9 + y^2/25 = 1$ about the $x$-axis.
Exercise 13 : Centroid of a Quarter Ellipse
Find the position of the centroid of the area of one quarter of an ellipse. The equation of the ellipse is $x^2/a^2 + y^2/b^2 = 1$.
Exercise 14 : Centroid of a Circular Sector
A plate is cut into a circular sector of $375 mm$ radius and $65^\circ$ included angle. Find the position of the centroid along the axis of symmetry.
Exercise 15 : Center of Mass with Radial Density Variation
The density of the material of which a right circular cone is made varies as the square of the distance from the vertex. Find the position of the center of mass.
Exercise 16 : Centroid of a Hemisphere
A hemisphere has a radius of $125mm$. Calculate the position of its centroid.
Exercise 17 : Moments of Inertia of a Cylindrical Shell
A cylindrical shell has a mass $M$, a radius $R$ and a length $L$. Calculate its moment of inertia about
- a central axis
- an axis about a diameter at one end
- an axis through its centroid and along a diameter
Exercise 18 : Moment of Inertia of a Steel Rod
A steel rod is $3.75m$ long and has a circular cross-section of $35mm$ diameter. The density of steel is $7800 \frac{kg}{m}^3$. Calculate the moment of inertia about:
- the centroid
- one end
Exercise 19 : Moment of Inertia of a Solid Cone
A solid right circular cone has a mass of $165kg$, a base radius of $175mm$, and a height of $650mm$. Calculate its moment of inertia about a central axis.
Exercise 20 : Hydrostatic Force on a Triangular Plate
A triangular plate of base 5 m and height $8m$ is immersed in a lake with its base along the water level. Calculate the total pressure on the plate and the depth of the center of pressure if the plate is vertical. Density of water $\rho_{W}= 1000 \frac{kg}{m}^3$
Exercise 21 : Work Done by a Variable Force
Calculate the work done in stretching a spring whose force obeys Hooke’s law with constant $k$, from an initial extension $x_1$ to a final extension $x_2$. Then generalize to a non-linear spring with force $F(x)=ax+bx^3$.
Exercise 22 : Volume by Pappus’ Theorem (Torus)
Using Pappus’ centroid theorem, find the volume of the torus generated by revolving a circle of radius $r$ whose centre is at a distance $R(>r)$ from the axis of revolution.
Exercise 23 : Hydrostatic Force on a Circular Plate
Find the total hydrostatic force on a vertical circular plate of radius $a$ whose centre is submerged at depth $h$ (measured to the centre). Determine also the centre of pressure.
Exercise 24 : Arc Length and Curvature of a Cycloid Segment
Find the arc length and curvature of the cycloid $x=\theta-\sin\theta$, $y=1-\cos\theta$ between $\theta=0$ and $\theta=2\pi$ and identify any cusps.
Exercise 25 : Surface Area of a Catenary of Revolution
Find the surface area generated by revolving the catenary $y=a\cosh(x/a)$ about the $x$-axis between two given ordinates $x=\pm b$.
Exercise 26 : Polar Area of a Cardioid
Calculate the area enclosed by the cardioid $r = a(1+\cos\theta)$.
Exercise 27 : Numerical Integration — Simpson’s Rule Application
Use Simpson’s rule with $n=4$ to approximate $\int_0^2 \frac{1}{1+x^2},dx$. Compare the result with the exact value.
Exercise 28 : Centroid of a Composite Plate
Find the centroid of the plane region bounded by $y=x^2$ and $y=4$ (between their intersections). Treat as a composite area and compute coordinates of the centroid.
Exercise 29 : Improper Integrals — Convergence and Evaluation
Investigate the convergence of $\int_1^{\infty} \frac{1}{x^p},dx$ and evaluate it for $p>1$. Also determine whether $\int_0^1 \frac{1}{x^p},dx$ converges and for which $p$.