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Chapter 5: Problem 61

Let $$ f(x)=a x^{3}+b x^{2}+c x+d $$ where \(a, b, c\), and \(d\) are constants. a) Compute \(\int_{-h}^{h} f(x) d x\). Your answer will contain the constants \(a, b, c, d\), and \(h\). b) Substitute \(x=h\) into equation (3) to find an expression for \(f(h)\) in terms of \(a, b, c, d\), and \(h .\) Do the same to find an expression for \(f(0)\) and \(f(-h)\) c) Use parts (a) and (b) to show that \(\int_{-h}^{h} f(x) d x=\frac{h}{3}[f(-h)+4 f(0)+f(h)]\) d) Explain why part (c) implies that Simpson's rule gives the exact answer for cubic polynomials.

Short Answer

Expert verified
The integral of a cubic polynomial over symmetric limits can be exactly computed using Simpson's rule, demonstrated by matching results of both methods.

Step by step solution

01

- Integrate the given function

Given the function \[ f(x) = ax^3 + bx^2 + cx + d \]. Compute the definite integral as follows:\[ \int_{-h}^{h} f(x) \, dx = \int_{-h}^{h} (ax^3 + bx^2 + cx + d) \, dx \].Split the integral into separate integrals:\[ \int_{-h}^{h} ax^3 \, dx + \int_{-h}^{h} bx^2 \, dx + \int_{-h}^{h} cx \, dx + \int_{-h}^{h} d \, dx \].Now compute each integral.
02

- Compute individual integrals

Integrate each term:\[ \int_{-h}^{h} ax^3 \, dx = a \left[ \frac{x^4}{4} \right]_{-h}^{h} \]. Since \(x^4\) is an even function, the contribution from \(ax^3\) over symmetric limits cancels out, resulting in 0.\[ \int_{-h}^{h} bx^2 \, dx = b \left[ \frac{x^3}{3} \right]_{-h}^{h} = b \left( \frac{h^3}{3} - \frac{(-h)^3}{3} \right) = b \left( \frac{h^3}{3} + \frac{h^3}{3} \right) = \frac{2bh^3}{3} \].\[ \int_{-h}^{h} cx \, dx = c \left[ \frac{x^2}{2} \right]_{-h}^{h} \]. Since \(x^2\) is an even function, the contribution from \(cx\) over symmetric limits cancels out, resulting in 0.\[ \int_{-h}^{h} d \, dx = d \left[ x \right]_{-h}^{h} = d(h - (-h)) = 2dh \].
03

- Combine results of the integrals

Combine the results of each integral:\[ \int_{-h}^{h} f(x) \, dx = 0 + \frac{2bh^3}{3} + 0 + 2dh = \frac{2bh^3}{3} + 2dh \].
04

- Evaluate the function at specific points

Evaluate the function at \(x = h\), \(x = 0\), and \(x = -h\):\[ f(h) = ah^3 + bh^2 + ch + d \].\[ f(0) = a(0)^3 + b(0)^2 + c(0) + d = d \]. \[ f(-h) = a(-h)^3 + b(-h)^2 + c(-h) + d = -ah^3 + bh^2 - ch + d \].
05

- Verify the given equality

Use the expressions derived:Left-hand side:\[ \int_{-h}^{h} f(x) \, dx = \frac{2bh^3}{3} + 2dh \].Right-hand side using Simpson's rule formula:\[ \frac{h}{3} [f(-h) + 4f(0) + f(h)] \].Substitute the expressions into the formula:\[ \frac{h}{3} [ -ah^3 + bh^2 - ch + d + 4d + ah^3 + bh^2 + ch + d ] = \frac{h}{3} [2bh^2 + 6d] = \frac{h}{3} \cdot 2h^2 b + h 6d = \frac{2bh^3}{3} + 2dh \].This matches the left-hand side, hence verifying the equality.
06

- Explain Simpson's rule implication

Since Simpson's rule computed exactly matches the integral results for a cubic polynomial, this shows that Simpson's rule gives the exact result for integrals of cubic polynomials.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Definite Integrals
A definite integral represents the net area under a curve from one point to another. For a function defined as f(x), the definite integral from point a to point b is represented by: \[ \int_{a}^{b} f(x) \, dx \] \
This calculation sums up all the tiny areas under the curve of f(x) from x = a to x = b. It's particularly useful in finding the total value, such as the total distance traveled given a speed function or the overall quantity from a rate function.

In our exercise, we are working with the function \[ f(x) = ax^3 + bx^2 + cx + d \] between -h and h. To find \[ \int_{-h}^{h} f(x) \, dx, \] we need to integrate each term separately. Let's break it down:
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