Problem 43 - Thomas Calculus

Question

Find the volume of the solid whose base is the region in the \(x y-\) plane that is bounded by the parabola \(y=4-x^{2}\) and the line \(y=3 x,\) while the top of the solid is bounded by the plane \(z=x+4 .\)

Step-by-Step Solution

Answer

The volume of the solid is 855 cubic units.

Step 1: Find Intersection Points

First, determine where the curves \(y=4-x^2\) and \(y=3x\) intersect by setting the equations equal to each other: \(4-x^2 = 3x\). Rearrange this as \(x^2 + 3x - 4 = 0\). Solve for \(x\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=3\), \(c=-4\). This yields the solutions \(x=1\) and \(x=-4\).

Step 2: Set Up Integral for Volume

The volume of the solid can be computed by integrating over the base region. For each slice at position \(x\), the height of the solid is determined by \(z = x + 4\). The bounds of the base are: \(-4 \leq x \leq 1\), and the region between the functions: \(3x \leq y \leq 4-x^2\).

Step 3: Evaluate Double Integral

Evaluate the double integral \(V = \int_{-4}^{1} \int_{3x}^{4-x^2} (x + 4) \; dy \, dx\). Start with the inner integral: \(\int_{3x}^{4-x^2} (x+4) \; dy = [(x+4)y]_{3x}^{4-x^2} = (x+4)(4-x^2 - 3x)\). Simplify it: \((4-x^2-3x)(x+4)\) and expand: \(4x+16 - x^3 - 3x^2 - 4x -12x - 3x -12\).

Step 4: Simplify and Integrate Over x

Combine like terms from the previous step to simplify the integrand: \(-x^3 - 3x^2 - 11x + 16\). Integrate over \([-4,1]\): \(\int_{-4}^{1} -x^3 - 3x^2 - 11x + 16 \; dx = [(-1/4)x^4 - x^3 - (11/2)x^2 + 16x]_{-4}^{1}\).

Step 5: Calculate the Final Volume

Calculate the definite integral by substituting the bounds \(1\) and \(-4\) into the antiderivative. Compute: \{(-1/4)(1)^4 - (1)^3 - (11/2)(1)^2 + 16(1)\} - \{(-1/4)(-4)^4 - (-4)^3 - (11/2)(-4)^2 + 16(-4)\}. This evaluates to \{0.25 -1 - 5.5 + 16\} - \{-64 - 64 - 88 - 64\}. Add and subtract the values to find \(3420/4=855\).

Key Concepts

Double Integration

Double integration is a powerful mathematical technique used to calculate the volume of solid regions over a defined planar area. In other words, it's like summing up small slices of a surface to find the total volume it covers. In this exercise, double integration helps us calculate the volume of a three-dimensional solid that's above a certain area in the xy-plane.

For problems involving double integration, you follow these steps:

• Set up the limits of integration based on the region of interest.
• Establish the height function for the solid.
• First, integrate with respect to one variable (say, y), while treating other variables as constants.
• Then, integrate the resulting expression with respect to the second variable (x).

By following these steps, you can determine the volume of a solid bound by specific surfaces and planes.

Parabolic Region

When discussing the volume of solids, it's important to identify the base region of the solid, often bound by curves or lines in the xy-plane. For this exercise, the base region is bound by a parabola and a line. Specifically, we have:

• The parabola defined by the equation: \( y = 4 - x^2 \)
• The line defined by the equation: \( y = 3x \)

These two curves intersect at points where their equations are equal, giving us intersection coordinates to determine the limits for integration. Solving \(y = 4 - x^2\) and \(y = 3x\) simultaneously gives us the x-intersections, crucial for setting the starting and ending limits for our integrations.The region these curves enclose forms the footprint of the solid whose volume we are computing.

Solid Geometry

Solid geometry involves understanding 3D shapes and structures formed from bounded surfaces. In this problem, we examine the solid formed above a parabolic region with its top defined by the plane \(z = x + 4\).

Here's how it works:

• The base is described by a region in the xy-plane, where the parabola and line define its boundaries.
• The height at any point \((x, y)\) in this region is determined by the function \(z = x + 4\), representing a tilted plane over the xy-plane.
• The volume is thus computed by considering how this plane extends over every point in the bounded region.

Understanding the interaction between the plane and the base region helps identify how large the solid is within these constraints.

Volume Integration

Volume integration is the holistic process by which we calculate the total space occupied by a 3D solid. In this case, volume integration melds together our understanding of geometry and calculus to solve the problem.

Steps for volume integration usually include:

• Setting up a double integral to contain both the area of the base and the function defining height.
• Successfully solving this double integral to find a numerical value, representing the total volume.
• Ensuring that the limits of all integrations accurately reflect the geometrical constraints.

Here, you apply the double integral \(V = \int \int \) to measure how high the plane \(z = x + 4\) rises above every point inside the parabolic base in the xy-plane. This gives us a structured approach to finding the volume by summing the contributions from all areas in the region.