Problem 57 - Calculus Early Transcendentals

Question

Let \(S\) be the solid in \(\mathbb{R}^{3}\) between the cylinder \(z=f(x)\) and the region \(R=\\{(x, y): a \leq x \leq b, c \leq y \leq d\\},\) where \(f(x) \geq 0\) on \(R .\) Explain why \(\int_{c}^{d} \int_{a}^{b} f(x) d x d y\) equals the area of the constant cross section of \(S\) multiplied by \((d-c),\) which is the volume of \(S\)

Step-by-Step Solution

Answer

Answer: Yes, the double integral ∫∫ f(x) dx dy over the region R equals the area of the constant cross-section of S multiplied by (d-c), which is the volume of the solid S.

Step 1: Compute the double integral of \(f(x)\) over the region \(R\)

The double integral to compute is \(\int_{c}^{d} \int_{a}^{b} f(x) dx dy\). To calculate the integral, first evaluate the inner integral with respect to \(x\) and then the outer integral with respect to \(y\). The integral can be given as: \(I = \int_{c}^{d} \left(\int_{a}^{b} f(x) dx\right) dy\)

Step 2: Calculate the area of the constant cross-section of S

The constant cross-section of \(S\) is given by the region where \(z = f(x)\). Since \(f(x)\) is non-negative on \(R\), the base of the solid is defined by the region \(R\). The cross-sectional area is given by: \(A = \int_{a}^b f(x) dx\)

Step 3: Multiply the area of the constant cross-section by (d-c)

To find the volume of \(S\), multiply the area of the constant cross-section by the height difference, which is \((d-c)\). The volume is given by: \(V = A(d-c)\) \(V = \left(\int_{a}^b f(x) dx\right)(d-c)\)

Step 4: Show that the results of the double integral and the volume are equal

We have found that the double integral \(I\) and the volume \(V\) can be represented as: \(I = \int_{c}^{d} \left(\int_{a}^{b} f(x) dx\right) dy\) \(V = \left(\int_{a}^b f(x) dx\right)(d-c)\) Now, we can rewrite the volume expression as a double integral as well: \(V = \int_{c}^{d} \left(\int_{a}^{b} f(x) dx\right) dy\) Comparing \(I\) and \(V\), we can conclude that they are equal: \(I = V\) Thus, the double integral \(\int_{c}^{d} \int_{a}^{b} f(x) dx dy\) equals the area of the constant cross-section of \(S\) multiplied by \((d-c)\), which is the volume of \(S\).