Problem 3 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

a. Write an integral that is the volume of the body with base the region of the \(x, y-\) plane bounded by $$ y_{1}=0.25 \sqrt{x} \sqrt[4]{2-x} \quad y_{2}=-0.25 \sqrt{x} \sqrt[4]{2-x} \quad 0 \leq x \leq 2 $$ and with each cross section perpendicular to the \(x\) -axis at \(x\) being a square with lower edge having endpoints \(\left[x, y_{2}(x), 0\right]\) and \(\left[x, y_{1}(x), 0\right]\) (see Exercise Figure 11.1.5A). (The value of the integral is \(4 \sqrt{2} / 15\) ). b. Write an integral that is the volume of the body with base the region of the \(\mathrm{x}, \mathrm{y}\) -plane bounded by $$ y_{1}=0.25 \sqrt{x} \sqrt[4]{2-x} \quad y_{2}=-0.25 \sqrt{x} \sqrt[4]{2-x} \quad 0 \leq x \leq 2 $$ and with each cross section perpendicular to the \(x\) -axis at \(x\) being an equilateral triangle with lower edge having endpoints \(\left[x, y_{2}(x), 0\right]\) and \(\left[x, y_{1}(x), 0\right]\) (see Exercise Figure \(11.1 .5 \mathrm{~B}\) ). (The value of the integral is \(\sqrt{6} / 15\) ).

Step-by-Step Solution

Answer

a. \(V = \frac{4\sqrt{2}}{15}\); b. \(V = \frac{\sqrt{6}}{15}\).

Step 1: Determine the side length of the cross-section

For both problems, the side of either a square or an equilateral triangle at any cross-section perpendicular to the x-axis spans from \(y_2(x)\) to \(y_1(x)\). Thus, the side length \(s\) is given by the expression: \[ s = y_1(x) - y_2(x) = 0.25 \sqrt{x} \sqrt[4]{2-x} - (-0.25 \sqrt{x} \sqrt[4]{2-x}) = 0.5 \sqrt{x} \sqrt[4]{2-x} \].

Step 2: Set up the integral for volume of squares

For problem (a), each cross-section is a square with side length \(s = 0.5 \sqrt{x} \sqrt[4]{2-x}\). The area of this square is \(s^2 = (0.5 \sqrt{x} \sqrt[4]{2-x})^2 = 0.25 x (\sqrt[4]{2-x})^2\). The volume of the solid is obtained by integrating this area along the interval \([0, 2]\): \[ V = \int_0^2 0.25 x (\sqrt[4]{2-x})^2 \, dx = \int_0^2 0.25 x (2-x)^{1/2} \, dx \].

Step 3: Set up the integral for volume of equilateral triangles

For problem (b), each cross-section is an equilateral triangle with side length \(s = 0.5 \sqrt{x} \sqrt[4]{2-x}\). The area of the triangle is \(\frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{4} (0.5 \sqrt{x} \sqrt[4]{2-x})^2 = \frac{\sqrt{3}}{16} x (\sqrt[4]{2-x})^2\). The volume of the solid is given by: \[ V = \int_0^2 \frac{\sqrt{3}}{16} x (2-x)^{1/2} \, dx \].

Step 4: Solve the integrals

For both integrals, perform the integration process to find the volumes. The evaluated integral for the square cross-sections results in the given volume of \(\frac{4\sqrt{2}}{15}\) and for the equilateral triangle cross-sections, the evaluation results in \(\frac{\sqrt{6}}{15}\). These values match the integral results as stated in the problem.

Key Concepts

Integral Calculus

Integral Calculus is a branch of mathematics focused on accumulation values and areas under curves. In this context, it involves finding the volume of a 3D shape using integration.
When you see an integral related to volume, it signifies how a series of shapes (in this case squares or equilateral triangles) stack to form a solid. This is accomplished by integrating the area of each shape across a given interval.
Each of these areas contributes to the total volume. The integral for volume can be thought of as summing infinite slices, precisely calculated, fitting within the bounds you are provided.

Volume Calculation

Calculating volume with integrals involves determining how space is enclosed within a solid.
In exercises like this, you start with a base area on the \(x, y\) plane, and each cross-section of the solid has a known shape—here, squares or equilateral triangles.
The volume of the square cross-sections is represented by the integral \[ V = \int_0^2 0.25x (2-x)^{1/2} \, dx \] and for the equilateral triangle, the volume is \[ V = \int_0^2 \frac{\sqrt{3}}{16}x (2-x)^{1/2} \, dx \]. These formulas represent how each slice contributes to the overall volume. As each cross-section varies across the interval defined by the bounds, the integral calculates this variation.

Cross-sections

Cross-sections are sections made of a 3D object, creating 2D shapes when sliced perpendicular to a certain axis—in this problem, perpendicular to the x-axis.
For this problem, each cross-section's size is determined by the functions \(y_1(x)\) and \(y_2(x)\), which give the upper and lower boundaries. This produces shapes that vary as \(x\) changes.
Understanding cross-sections helps in conceptualizing how integrals apply. By knowing the function boundaries, you can determine how width (side length for squares, edge for equilateral triangles) changes along the axis, calculating the area and resulting volume effectively.

• Square cross-sections simplify to side length squared.
• Equilateral triangles relate to side length using specific geometric formulas.