Mathematics Foundations/10.3 Applications of Integration

Applications of Integration Integration is one of the most powerful tools in mathematics, with applications spanning across numerous fields including physics, engineering, economics, and biology. This chapter explores the diverse applications of integration beyond mere antidifferentiation. Area Between Curves One of the fundamental applications of integration is finding the area between two curves. If we have two functions f(x) and g(x) where f(x) ≥ g(x) on an interval [a, b], the area between these curves is given by: Example 1 Find the area between the curves y = x² and y = x + 2 over the interval where they intersect. Solution: First, we find the points of intersection by solving: So the curves intersect at x = -1 and x = 2. To determine which function is greater in this interval, we note that at x = 0: When x = 0, x² = 0 and x + 2 = 2 So x + 2 > x² at x = 0 Therefore, in the interval [-1, 2], we have: Practice Problem 1 Find the area between the curves y = sin(x) and y = cos(x) over the interval [0, π/2]. Volumes of Revolution Another important application is finding volumes of three-dimensional objects formed by revolving a region around an axis. Disk Method When revolving a region around the x-axis, if the region is bounded by y = f(x), x = a, x = b, and the x-axis, the volume is: Washer Method If the region is bounded by two curves y = f(x) and y = g(x), where f(x) ≥ g(x) ≥ 0, the volume is: Shell Method When revolving around the y-axis, we use the shell method: Example 2 Find the volume of the solid formed by revolving the region bounded by y = x², y = 0, and x = 2 around the x-axis. Solution: Using the disk method: Practice Problem 2 Find the volume of the solid formed by revolving the region bounded by y = √x, y = 0, and x = 4 around the y-axis. Arc Length Integration can be used to find the length of a curve. For a function y = f(x) on [a, b], the arc length is: Example 3 Find the arc length of the curve y = x³/6 + 1/(2x) from x = 1 to x = 2. Solution: First, we find the derivative: Now we calculate: The arc length is: This simplifies to: Practice Problem 3 Find the arc length of the curve y = ln(sin(x)) from x = π/6 to x = π/3. Surface Area For a surface formed by revolving a curve y = f(x) around the x-axis, the surface area is: Example 4 Find the surface area of the cone formed by revolving the line y = x from x = 0 to x = h around the x-axis. Solution: For y = x, we have dy/dx = 1, so: Practice Problem 4 Find the surface area of the solid formed by revolving the curve y = x² from x = 0 to x = 1 around the x-axis. Work and Energy In physics, work done by a variable force F(x) over a distance from x = a to x = b is: Example 5 A spring has a natural length of 10 cm. If a force of 5 N is required to stretch it to 15 cm, find the work done in stretching it from 10 cm to 20 cm. Solution: By Hooke's Law, F = kx, where k is the spring constant. From the given information: The work done is: Practice Problem 5 A 5 kg object is lifted from the surface of the Earth to a height of 1000 m. Find the work done against gravity. (Use g = 9.8 m/s²) Center of Mass For a lamina with density function ρ(x,y), the center of mass is given by: For a uniform lamina bounded by y = f(x), y = 0, x = a, and x = b, these simplify to: Example 6 Find the center of mass of a uniform semicircular plate of radius R. Solution: Due to symmetry, we know that \bar{x} = 0. For \bar{y}: The denominator is the area of the semicircle: πR²/2. For the numerator: Therefore: Practice Problem 6 Find the center of mass of a uniform quarter-circular plate of radius R in the first quadrant. Fluid Pressure and Force The force exerted by a fluid on a vertical plate is: where ρ is the fluid density, g is gravitational acceleration, and h is the height of the fluid surface. Example 7 A triangular plate is submerged vertically in water with its top vertex at the water surface and its base, of length 4 m, parallel to the water surface at a depth of 3 m. Find the force exerted by the water on one side of the plate. (Use ρ = 1000 kg/m³ and g = 9.8 m/s²) Solution: We can set up coordinates with the origin at the top vertex, y-axis pointing downward. The triangular plate has vertices at (0,0), (-2,3), and (2,3). The equation of the left side is: for -2 ≤ x ≤ 0 The equation of the right side is: for 0 ≤ x ≤ 2 The force is: where y(x) represents the boundary of the triangle. This evaluates to: Practice Problem 7 A semicircular plate of radius 2 m is submerged vertically in water with its diameter along the water surface. Find the force exerted by the water on one side of the plate. Probability and Expected Value For a continuous random variable X with probability density function f(x), the expected value is: Example 8 A random variable X has probability density function f(x) = 2x for 0 ≤ x ≤ 1 and f(x) = 0 elsewhere. Find E[X]. Solution: Practice Problem 8 A random variable X has probability density function f(x) = 3x² for 0 ≤ x ≤ 1 and f(x) = 0 elsewhere. Find E[X]. Economics Applications In economics, integration is used to calculate consumer and producer surplus, total revenue, and other economic metrics. Example 9 The marginal cost function for producing x units of a product is MC(x) = 20 + 0.1x². If the fixed costs are $500, find the total cost of producing 100 units. Solution: Total cost = Fixed costs + Variable costs Variable costs = ∫ MC(x) dx Practice Problem 9 The marginal revenue function for a product is MR(x) = 50 - 0.2x. Find the total revenue from selling 100 units. Exercises 1. Find the area between the curves y = e^x and y = e^(-x) from x = 0 to x = 1. 2. A solid is formed by revolving the region bounded by y = x^(1/3), y = 0, and x = 8 around the y-axis. Find its volume. 3. Find the arc length of the curve y = ln(cos(x)) from x = 0 to x = π/4. 4. A tank has the shape of a hemisphere with radius 5 m. If it is filled with water to a height of 3 m, find the volume of water in the tank. 5. Find the center of mass of a uniform lamina bounded by y = x^2 and y = 4. 6. A rectangular plate with width 2 m and height 3 m is submerged vertically in water with its top edge 1 m below the water surface. Find the force exerted by the water on one side of the plate. 7. The probability density function of a random variable X is f(x) = kx(1-x) for 0 ≤ x ≤ 1 and f(x) = 0 elsewhere. Find the value of k and then calculate E[X]. 8. A company's marginal profit function is MP(x) = 100 - 0.2x - 0.001x^2, where x is the number of units produced. Find the change in profit when production increases from 100 to 200 units. 9. Find the volume of the solid formed by revolving the region bounded by y = sin(x), y = 0, x = 0, and x = π around the x-axis. 10. A particle moves along a straight line with velocity v(t) = t^2 - 4t + 3 m/s. Find the total distance traveled by the particle during the first 5 seconds.