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Magazine Chapter 9: Problem 15
Approximate the following integrals by the midpoint rule, the trapezoidal rule, and Simpson's rule. Then, find the exact value by integration. Express your answers to five decimal places. $$\int_{1}^{4}(2 x-3)^{3} d x ; n=3$$
Short Answer
Expert verified
Midpoint rule: 23.87500, Trapezoidal rule: 88.83333, Simpson's rule: 89.58333, Exact value: 511.875.
Step by step solution
01
- Determine the Interval and Subintervals
The integral is from 1 to 4 and we are approximating this with n=3 subintervals. Calculate the width of each subinterval: \[ \text{Width} \triangle x = \frac{4-1}{3} = 1 \text{ unit} \].
02
- Midpoint Rule
For the midpoint rule, we evaluate the function at the midpoint of each subinterval. The points are: \[\begin{align*} x_1 &= 1.5, \ x_2 &= 2.5, \ x_3 &= 3.5\end{align*}\].The function values at these points are: \[\begin{align*}(2(1.5)-3)^3 &= (-0.5)^3 = -0.125, \ (2(2.5)-3)^3 &= (2)^3 = 8, \ (2(3.5)-3)^3 &= (4)^3 = 64\end{align*}\].Approximate the integral: \[ \text{Midpoint Approximation} = 1 \times \frac{1}{3}(-0.125 + 8 + 64) = 1 \times 23.29167 = 23.87500 \].
03
- Trapezoidal Rule
For the trapezoidal rule, evaluate the function at the endpoints of each subinterval. The points are: \[\begin{align*}x_0 &= 1, \ x_1 &= 2, \ x_2 &= 3, \ x_3 &= 4\end{align*} \].The function values at these points are: \[\begin{align*} (2(1)-3)^3 &= (-1)^3 = -1, \ (2(2)-3)^3 &= (1)^3 = 1, \ (2(3)-3)^3 &= (3)^3 = 27, \ (2(4)-3)^3 &= (5)^3 = 125\end{align*} \].Approximate the integral: \[ \text{Trapezoidal Approximation} = \frac{1}{2} \times 1 \times \big( (-1) + 2(1 + 27) + 125 \big) = \frac{1}{2} \times ( 1 + 54 + 125 ) = 88.83333 \].
04
- Simpson's Rule
Simpson's rule uses parabolic arcs to approximate the function. Use the points: \[\begin{align*} x_0 &= 1, \ x_1 &= 2, \ x_2 &= 3, \ x_3 &= 4 \end{align*} \].The function values at these points are: \[\begin{align*} f(x_0) &= -1, \ f(x_1) &= 1, \ f(x_2) &= 27, \ f(x_3) &= 125 \end{align*} \].Approximate the integral: \[ \text{Simpson's Approximation} = \frac{1}{3} \times \frac{1}{3} \times \big( (-1) + 4(1) + 2(27) + 4(125) + (-1) \big)= \frac{1}{3} \times ( -1 + 4 + 54 + 500 + 125-1 ) = 89.58333 \].
05
- Exact Value by Integration
Evaluate the definite integral by finding the antiderivative and then computing the difference: \[\begin{align*} \text{Let } u &= 2x-3, \ f(u) &= u^3 \end{align*} \] so \[\text{ du = 2dx } \]. Hence we have, \[ \frac{1}{2} \times \bigg[ \frac{(2x-3)^4}{4} \bigg]_{1}^{4} \] Computing this with definite intervals, \[ \frac{1}{2} \times \bigg( \frac{1}{4} \times ((8)^4 - (-1)^4) \bigg) \ = \frac{1}{8} (4096 - 1) = 511.87500 \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Midpoint Rule
The Midpoint Rule is a numerical method for approximating the value of a definite integral. This rule leverages the value of the function at the midpoint of subintervals, providing a quick estimate of the area under a curve.
To apply the Midpoint Rule, follow these steps:
To apply the Midpoint Rule, follow these steps:
- Divide the interval \[a, b\] into n subintervals of equal width. The width of each subinterval is given by \[ \Delta x = \frac{b-a}{n} \].
- Calculate the midpoint of each subinterval. If the interval is from 1 to 4 and \[ n=3 \], then the midpoints are at 1.5, 2.5, and 3.5.
- Evaluate the function at each midpoint. For example, if the function is \[ f(x) = (2x-3)^3 \], calculate these values for each midpoint.
- Sum these values, multiply by the width of the subinterval, and you get the Midpoint Approximation of the integral.
Trapezoidal Rule
The Trapezoidal Rule approximates the area under a curve using trapezoids rather than rectangles like in the Midpoint Rule.
Here's how to apply the Trapezoidal Rule:
Here's how to apply the Trapezoidal Rule:
- Divide the interval \[a, b\] into n subintervals of equal width. Compute the width as \[ \Delta x = \frac{b-a}{n} \].
- Evaluate the function at the endpoints of these subintervals. For instance, if \[ f(x) = (2x-3)^3 \], you would calculate the values of the function at 1, 2, 3, and 4.
- Use these endpoint values to calculate the sum: \[ \text{Trapezoidal Approximation} = \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + f(x_n) \right] \].
Simpson's Rule
For even better accuracy, use Simpson鈥檚 Rule. It approximates the definite integral by fitting parabolas to the curve rather than using straight lines.
To use Simpson's Rule:
To use Simpson's Rule:
- Subdivide the interval \[a, b\] into an even number of subintervals (n must be even), each of width \[ \Delta x = \frac{b-a}{n} \].
- Calculate the function values at each subinterval mark: \[ f(x_0), f(x_1), \ldots, f(x_n) \].
- Apply the Simpson鈥檚 Rule formula: \[ \text{Simpson's Approximation} = \frac{\Delta x}{3} \left[ f(x_0) + 4 \sum_{i=1,3,5,\dots}^{n-1} f(x_i) + 2f(x_{2,4,6,\dots}^{n-2}) + f(x_n) \right] \], where the odd indexes are multiplied by 4 and the even ones (excluding the endpoints) by 2.
Definite Integral
The Definite Integral of a function over an interval \[a, b\] represents the accumulation of quantities, which can be thought of as the net area under the curve between these points.
To understand a definite integral, consider the integral:
\[ \int_{1}^{4} (2x-3)^3 dx \].
This involves finding an antiderivative F(x) of the function and then computing the difference F(b) - F(a), where F(x) is such that F'(x) = (2x-3)^3.
Definite integrals give the exact value, unlike numerical methods like the Midpoint, Trapezoidal, or Simpson's Rules, which only provide approximations.
To understand a definite integral, consider the integral:
\[ \int_{1}^{4} (2x-3)^3 dx \].
This involves finding an antiderivative F(x) of the function and then computing the difference F(b) - F(a), where F(x) is such that F'(x) = (2x-3)^3.
Definite integrals give the exact value, unlike numerical methods like the Midpoint, Trapezoidal, or Simpson's Rules, which only provide approximations.
Antiderivative
An Antiderivative of a function f(x) is another function F(x) such that F'(x) = f(x). Antiderivatives are crucial in finding the exact value of definite integrals.
To find the antiderivative of a given function:
To find the antiderivative of a given function:
- Identify the function whose derivative matches the given function.
- In our example, for \[ f(x) = (2x-3)^3 \], we can use a substitution method where we let \[ u = 2x - 3 \]. Then, \[ du = 2dx \], and the integral becomes easier to solve.
- Integrate the substituted function and then reverse the substitution.
- Inserting the limits of integration, calculate F(b) - F(a).
