Problem 164
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. $$ y=x^{2} \text { over }[0,4] $$
Step-by-Step Solution
Verified Answer
The exact area under the curve is 21.33.
1Step 1: Define the Function and Interval
We are given the function \(y = x^2\) over the interval \([0, 4]\). Our task is to estimate and then determine exactly the area under this curve over the given interval.
2Step 2: Calculate the Width of Each Rectangle
For the Riemann sums, divide the interval \([0, 4]\) into \(N = 10\) equal parts. The width of each rectangle, \(\Delta x\), is \(\frac{4 - 0}{10} = 0.4\).
3Step 3: Calculate Left-Endpoint Riemann Sum
To find the left-endpoint Riemann sum \(L_{10}\), evaluate the function at each left endpoint of the intervals: \(x = 0, 0.4, 0.8, ..., 3.6\). Sum these values: \[L_{10} = 0.4 (f(0) + f(0.4) + \, \ldots \, + f(3.6) ) = 0.4(0 + 0.16 + 0.64 + \, \ldots \, + 12.96)\].
4Step 4: Calculate Right-Endpoint Riemann Sum
Similarly, for the right-endpoint Riemann sum \(R_{10}\), evaluate the function at each right endpoint: \(x = 0.4, 0.8, ..., 4\). Sum these values: \[R_{10} = 0.4 (f(0.4) + f(0.8) + \, \ldots \, + f(4) ) = 0.4(0.16 + 0.64 + \. \, \ldots \, + 16)\].
5Step 5: Calculate the Average of Riemann Sums (\(T_{10}\))
Calculate \(T_{10}\), the average of the left and right sums:\[T_{10} = \frac{L_{10} + R_{10}}{2}\].
6Step 6: Apply the Fundamental Theorem of Calculus
To find the exact area, use the Fundamental Theorem of Calculus:\[\int_{0}^{4} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{4} = \frac{4^3}{3} - \frac{0^3}{3} = \frac{64}{3} = 21.33 \].
7Step 7: Compare the Estimates and Exact Area
Compare the average \(T_{10}\) with the exact value \(21.33\). Both values should be close, with \(T_{10}\) providing an approximation of the integral.
Key Concepts
Fundamental Theorem of CalculusLeft-Endpoint Riemann SumRight-Endpoint Riemann Sum
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus connects differentiation and integration, two central concepts in calculus. It is divided into two parts, and we are particularly interested in Part 2 for this problem.
Part 2 of the Fundamental Theorem of Calculus is extremely useful because it allows us to compute definite integrals. When we need to find the exact area under a curve, as we do in this exercise, this theorem provides us with a method.
For the function given, which is \(y=x^2\), the theorem tells us to find an antiderivative of \(y\). In this case, a suitable antiderivative is \(F(x) = \frac{x^3}{3}\). Using the boundaries of the interval \([0, 4]\), the theorem lets us evaluate the definite integral as \(\int_{0}^{4} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{4} = \frac{4^3}{3} - \frac{0^3}{3} = \frac{64}{3} = 21.33\).
In simplistic terms, this means we evaluate the antiderivative at the upper boundary, subtract the evaluation at the lower boundary, and this results in the area under the curve over the interval.
Part 2 of the Fundamental Theorem of Calculus is extremely useful because it allows us to compute definite integrals. When we need to find the exact area under a curve, as we do in this exercise, this theorem provides us with a method.
For the function given, which is \(y=x^2\), the theorem tells us to find an antiderivative of \(y\). In this case, a suitable antiderivative is \(F(x) = \frac{x^3}{3}\). Using the boundaries of the interval \([0, 4]\), the theorem lets us evaluate the definite integral as \(\int_{0}^{4} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{4} = \frac{4^3}{3} - \frac{0^3}{3} = \frac{64}{3} = 21.33\).
In simplistic terms, this means we evaluate the antiderivative at the upper boundary, subtract the evaluation at the lower boundary, and this results in the area under the curve over the interval.
Left-Endpoint Riemann Sum
The Left-Endpoint Riemann Sum is an approximation method for definite integrals. It is a means to estimate the area under a curve using rectangles, which is particularly helpful when calculating exact area is not possible or practical.
To calculate the Left-Endpoint Riemann Sum, we use rectangles whose heights are determined by the function value at the left endpoints of subintervals. In this particular exercise, the interval \([0,4]\) is divided into 10 subintervals, each of width \(\Delta x = 0.4\). The left endpoints are \(0, 0.4, 0.8, \ldots, 3.6\).
The height of each rectangle corresponds to the function values \(f(0), f(0.4), \ldots, f(3.6)\). By summing these heights and multiplying by \(\Delta x\), we find the Left-Endpoint Riemann Sum, \(L_{10}\). This sum represents an estimate of the integral's value. While not exact, it gives us a reasonable approximation: \(L_{10} = 0.4(0 + 0.16 + 0.64 + ... + 12.96)\).
To calculate the Left-Endpoint Riemann Sum, we use rectangles whose heights are determined by the function value at the left endpoints of subintervals. In this particular exercise, the interval \([0,4]\) is divided into 10 subintervals, each of width \(\Delta x = 0.4\). The left endpoints are \(0, 0.4, 0.8, \ldots, 3.6\).
The height of each rectangle corresponds to the function values \(f(0), f(0.4), \ldots, f(3.6)\). By summing these heights and multiplying by \(\Delta x\), we find the Left-Endpoint Riemann Sum, \(L_{10}\). This sum represents an estimate of the integral's value. While not exact, it gives us a reasonable approximation: \(L_{10} = 0.4(0 + 0.16 + 0.64 + ... + 12.96)\).
Right-Endpoint Riemann Sum
Like its left-endpoint counterpart, the Right-Endpoint Riemann Sum is used to approximate the area under a curve by using rectangles. However, in this method, the height of each rectangle is determined by the function value at the right endpoints of each subinterval.
In our case, the same interval \([0,4]\) is divided into 10 subintervals, resulting in a width of \(\Delta x = 0.4\). The right endpoints that determine the rectangle heights are \(0.4, 0.8, 1.2, \ldots, 4\).
By evaluating the function \(f(x) = x^2\) at each of these right endpoints, we get the set of values \(f(0.4), f(0.8), ..., f(4)\). When these values are summed up and multiplied by \(\Delta x\), we arrive at the Right-Endpoint Riemann Sum, \(R_{10}\).
This sum offers another approximation of the definite integral, providing a glimpse into the behavior and changes of the function across the specified interval: \(R_{10} = 0.4(0.16 + 0.64 + ... + 16)\). The precise nature of this sum can vary somewhat when compared to the left-endpoint method, but together they help form an overall picture of integration.
In our case, the same interval \([0,4]\) is divided into 10 subintervals, resulting in a width of \(\Delta x = 0.4\). The right endpoints that determine the rectangle heights are \(0.4, 0.8, 1.2, \ldots, 4\).
By evaluating the function \(f(x) = x^2\) at each of these right endpoints, we get the set of values \(f(0.4), f(0.8), ..., f(4)\). When these values are summed up and multiplied by \(\Delta x\), we arrive at the Right-Endpoint Riemann Sum, \(R_{10}\).
This sum offers another approximation of the definite integral, providing a glimpse into the behavior and changes of the function across the specified interval: \(R_{10} = 0.4(0.16 + 0.64 + ... + 16)\). The precise nature of this sum can vary somewhat when compared to the left-endpoint method, but together they help form an overall picture of integration.
Other exercises in this chapter
Problem 157
In the following exercises, use the Fundamental Theorem of Calculus, Part \(1,\) to find each derivative. $$ \frac{d}{d x} \int_{1}^{x^{2}} \frac{\sqrt{t}}{1+t}
View solution 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 165
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