Problem 165
Question
In the following exercises, use a calculator to estimate the area under the curve by computing \(T_{10}\) , the average of the left- and right-endpoint Riemann sums using \(N=10\) rectangles. Then, using the Fundamental Theorem of Calculus, Part 2 , determine the exact area. $$ y=x^{3}+6 x^{2}+x-5 \text { over }[-4,2] $$
Step-by-Step Solution
Verified Answer
The estimated area using Riemann sums is approximately 88, and the exact area using the FTC is 88.
1Step 1: Define the Function and Interval
The function given is \( y = x^3 + 6x^2 + x - 5 \) over the interval \([-4, 2]\). We need to estimate the area under this curve within this interval using Riemann sums.
2Step 2: Calculate the Width of Each Rectangle
To use Riemann sums, we first calculate the width of each rectangle. The interval is from \(-4\) to \(2\), so it has a length of \(2 - (-4) = 6\). Given that \(N = 10\) rectangles are used, the width \(\Delta x\) of each rectangle is \( \Delta x = 6/10 = 0.6 \).
3Step 3: Calculate the Left-Endpoint Sum
For the left-endpoint Riemann sum, we evaluate the function at points starting at the left end \(-4\) with \(\Delta x = 0.6\) increments. The points are \(-4, -3.4, -2.8, ..., 1.4\). Calculate the sum: \[ L = \sum_{i=0}^{9} f(-4 + i\cdot0.6) \cdot 0.6 \]
4Step 4: Calculate the Right-Endpoint Sum
For the right-endpoint Riemann sum, we evaluate the function starting from \(-3.4\) to \(2\) (the right end). The points are \(-3.4, -2.8, ..., 2\). Calculate the sum: \[ R = \sum_{i=1}^{10} f(-4 + i\cdot0.6) \cdot 0.6 \]
5Step 5: Average the Left and Right Sums
The trapezoidal approximation \( T_{10} \) is the average of the left and right endpoint sums: \[ T_{10} = \frac{L + R}{2} \] This gives an estimate of the area under the curve over the interval \([-4, 2]\).
6Step 6: Compute Exact Area Using FTC Part 2
Use the Fundamental Theorem of Calculus Part 2. The antiderivative \( F(x) \) of \( f(x) = x^3 + 6x^2 + x - 5 \) is \( F(x) = \frac{x^4}{4} + 2x^3 + \frac{x^2}{2} - 5x + C \). Compute \( F(2) - F(-4) \): \[ F(2) = \frac{2^4}{4} + 2 \cdot 2^3 + \frac{2^2}{2} - 5 \cdot 2 = 4 + 16 + 2 - 10 = 12 \] \[ F(-4) = \frac{(-4)^4}{4} + 2 \cdot (-4)^3 + \frac{(-4)^2}{2} - 5 \cdot (-4) = 64/4 - 128 + 8 + 20 = -76 \] The exact area is \( F(2) - F(-4) = 12 - (-76) = 88 \).
Key Concepts
Fundamental Theorem of CalculusTrapezoidal ApproximationAntiderivatives
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus (FTC) is a bridge between differentiation and integration, two core concepts in calculus. It has two main parts. In this explanation, we focus on Part 2, which deals with evaluating definite integrals.
FTC Part 2 states that if you can find an antiderivative of a function, then you can compute the definite integral of that function over a specific interval simply by evaluating the antiderivative at the endpoints of that interval.
This can be written as: \[\int_{a}^{b} f(x) \, dx = F(b) - F(a)\]where \(F(x)\) is an antiderivative of \(f(x)\). In our problem, this theorem helps find the exact area under the curve \(y = x^3 + 6x^2 + x - 5\) from \(-4\) to \(2\).
By substituting the endpoints into the antiderivative function and subtracting \(F(-4)\) from \(F(2)\), we accurately determine the area under the curve, bypassing the approximation methods like Riemann sums.
FTC Part 2 states that if you can find an antiderivative of a function, then you can compute the definite integral of that function over a specific interval simply by evaluating the antiderivative at the endpoints of that interval.
This can be written as: \[\int_{a}^{b} f(x) \, dx = F(b) - F(a)\]where \(F(x)\) is an antiderivative of \(f(x)\). In our problem, this theorem helps find the exact area under the curve \(y = x^3 + 6x^2 + x - 5\) from \(-4\) to \(2\).
By substituting the endpoints into the antiderivative function and subtracting \(F(-4)\) from \(F(2)\), we accurately determine the area under the curve, bypassing the approximation methods like Riemann sums.
Trapezoidal Approximation
Trapezoidal approximation is a technique used to estimate the area under a curve. It's often more accurate than simple left or right Riemann sums.
In our exercise, we use the average of left and right-endpoint Riemann sums to calculate what is called \(T_{10}\), representing the trapezoidal approximation with 10 subintervals.
Each subinterval is considered as a trapezoid rather than a rectangle, hence improving the estimate by accounting for the slope of the function between points. This method is efficient when a quick estimate of an integral is necessary and computing the exact solution via antiderivatives is impractical.
In our exercise, we use the average of left and right-endpoint Riemann sums to calculate what is called \(T_{10}\), representing the trapezoidal approximation with 10 subintervals.
Each subinterval is considered as a trapezoid rather than a rectangle, hence improving the estimate by accounting for the slope of the function between points. This method is efficient when a quick estimate of an integral is necessary and computing the exact solution via antiderivatives is impractical.
- First, compute the left-endpoint sum, \(L\).
- Next, compute the right-endpoint sum, \(R\).
- Finally, calculate the average: \[T_{10} = \frac{L + R}{2}\]This value gives you an estimation of the area under the curve over the specified interval. While it's not perfectly precise, it usually comes very close, especially as the number of subintervals increases.
Antiderivatives
Antiderivatives, also called indefinite integrals, are functions that "reverse" differentiation. Finding an antiderivative means determining a function whose derivative yields the original function.
In equations, if \(F'(x) = f(x)\), then \(F(x)\) is an antiderivative of \(f(x)\). These forms are crucial when applying the Fundamental Theorem of Calculus for computing definite integrals.
In our example of \(f(x) = x^3 + 6x^2 + x - 5\), the antiderivative is \[F(x) = \frac{x^4}{4} + 2x^3 + \frac{x^2}{2} - 5x + C\]where \(C\) represents the constant of integration. Although \(C\) disappears in definite integral calculations, it's an important aspect in indefinite integrals.
Finding the correct antiderivative is essential for accurately computing areas under curves using FTC. Each term in the original function has its corresponding antiderivative term, revealing the cumulative effect gathered by the curve on the specified interval.
In equations, if \(F'(x) = f(x)\), then \(F(x)\) is an antiderivative of \(f(x)\). These forms are crucial when applying the Fundamental Theorem of Calculus for computing definite integrals.
In our example of \(f(x) = x^3 + 6x^2 + x - 5\), the antiderivative is \[F(x) = \frac{x^4}{4} + 2x^3 + \frac{x^2}{2} - 5x + C\]where \(C\) represents the constant of integration. Although \(C\) disappears in definite integral calculations, it's an important aspect in indefinite integrals.
Finding the correct antiderivative is essential for accurately computing areas under curves using FTC. Each term in the original function has its corresponding antiderivative term, revealing the cumulative effect gathered by the curve on the specified interval.
Other exercises in this chapter
Problem 158
In the following exercises, use the Fundamental Theorem of Calculus, Part \(1,\) to find each derivative. $$ \frac{d}{d x} \int_{0}^{\ln x} e^{t} d t $$
View solution Problem 164
In the following exercises, use a calculator to estimate the area under the curve by computing \(T_{10}\) , the average of the left- and right-endpoint Riemann
View solution Problem 166
In the following exercises, use a calculator to estimate the area under the curve by computing \(T_{10}\) , the average of the left- and right-endpoint Riemann
View solution Problem 167
In the following exercises, use a calculator to estimate the area under the curve by computing \(T_{10}\) , the average of the left- and right-endpoint Riemann
View solution