**Problem 7c** Use the table below to estimate \(\int_{0}^{40} f(x) \, dx\). Use a left-hand sum with \(n = 4\). \[ \begin{array}{c|ccccc} x & 0 & 10 & 20 & 30 & 40 \\ \hline f(x) & 350 & 410 & 435 & 450 & 460 \\ \end{array} \] **Explanation:** To estimate the integral \(\int_{0}^{40} f(x) \, dx\) using a left-hand sum with \(n = 4\), divide the interval \([0, 40]\) into 4 equal subintervals. Each subinterval will have a width of \(\Delta x = 10\). Since it's a left-hand sum, use the left endpoints of each subinterval to find the function values: - For the interval \([0, 10]\), use \(f(0) = 350\). - For the interval \([10, 20]\), use \(f(10) = 410\). - For the interval \([20, 30]\), use \(f(20) = 435\). - For the interval \([30, 40]\), use \(f(30) = 450\). The left-hand sum is given by: \[ \sum_{i=0}^{3} f(x_i) \Delta x = (350 + 410 + 435 + 450) \times 10 \] Calculate the sum: \[ 350 + 410 + 435 + 450 = 1645 \] Multiply by the width \(\Delta x = 10\): \[ 1645 \times 10 = 16450 \] Therefore, the estimated value of the integral is 16,450.
**Problem 7c** Use the table below to estimate \(\int_{0}^{40} f(x) \, dx\). Use a left-hand sum with \(n = 4\). \[ \begin{array}{c|ccccc} x & 0 & 10 & 20 & 30 & 40 \\ \hline f(x) & 350 & 410 & 435 & 450 & 460 \\ \end{array} \] **Explanation:** To estimate the integral \(\int_{0}^{40} f(x) \, dx\) using a left-hand sum with \(n = 4\), divide the interval \([0, 40]\) into 4 equal subintervals. Each subinterval will have a width of \(\Delta x = 10\). Since it's a left-hand sum, use the left endpoints of each subinterval to find the function values: - For the interval \([0, 10]\), use \(f(0) = 350\). - For the interval \([10, 20]\), use \(f(10) = 410\). - For the interval \([20, 30]\), use \(f(20) = 435\). - For the interval \([30, 40]\), use \(f(30) = 450\). The left-hand sum is given by: \[ \sum_{i=0}^{3} f(x_i) \Delta x = (350 + 410 + 435 + 450) \times 10 \] Calculate the sum: \[ 350 + 410 + 435 + 450 = 1645 \] Multiply by the width \(\Delta x = 10\): \[ 1645 \times 10 = 16450 \] Therefore, the estimated value of the integral is 16,450.
Calculus: Early Transcendentals
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ISBN:9781285741550
Author:James Stewart
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Chapter1: Functions And Models
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Question
![**Problem 7c**
Use the table below to estimate \(\int_{0}^{40} f(x) \, dx\). Use a left-hand sum with \(n = 4\).
\[
\begin{array}{c|ccccc}
x & 0 & 10 & 20 & 30 & 40 \\
\hline
f(x) & 350 & 410 & 435 & 450 & 460 \\
\end{array}
\]
**Explanation:**
To estimate the integral \(\int_{0}^{40} f(x) \, dx\) using a left-hand sum with \(n = 4\), divide the interval \([0, 40]\) into 4 equal subintervals. Each subinterval will have a width of \(\Delta x = 10\).
Since it's a left-hand sum, use the left endpoints of each subinterval to find the function values:
- For the interval \([0, 10]\), use \(f(0) = 350\).
- For the interval \([10, 20]\), use \(f(10) = 410\).
- For the interval \([20, 30]\), use \(f(20) = 435\).
- For the interval \([30, 40]\), use \(f(30) = 450\).
The left-hand sum is given by:
\[
\sum_{i=0}^{3} f(x_i) \Delta x = (350 + 410 + 435 + 450) \times 10
\]
Calculate the sum:
\[
350 + 410 + 435 + 450 = 1645
\]
Multiply by the width \(\Delta x = 10\):
\[
1645 \times 10 = 16450
\]
Therefore, the estimated value of the integral is 16,450.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F569dbbe0-fbf4-4ebf-ba85-ff97d3fc3240%2F0f5dc452-9588-40b8-a897-f2deeb517174%2Fhq90t68_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 7c**
Use the table below to estimate \(\int_{0}^{40} f(x) \, dx\). Use a left-hand sum with \(n = 4\).
\[
\begin{array}{c|ccccc}
x & 0 & 10 & 20 & 30 & 40 \\
\hline
f(x) & 350 & 410 & 435 & 450 & 460 \\
\end{array}
\]
**Explanation:**
To estimate the integral \(\int_{0}^{40} f(x) \, dx\) using a left-hand sum with \(n = 4\), divide the interval \([0, 40]\) into 4 equal subintervals. Each subinterval will have a width of \(\Delta x = 10\).
Since it's a left-hand sum, use the left endpoints of each subinterval to find the function values:
- For the interval \([0, 10]\), use \(f(0) = 350\).
- For the interval \([10, 20]\), use \(f(10) = 410\).
- For the interval \([20, 30]\), use \(f(20) = 435\).
- For the interval \([30, 40]\), use \(f(30) = 450\).
The left-hand sum is given by:
\[
\sum_{i=0}^{3} f(x_i) \Delta x = (350 + 410 + 435 + 450) \times 10
\]
Calculate the sum:
\[
350 + 410 + 435 + 450 = 1645
\]
Multiply by the width \(\Delta x = 10\):
\[
1645 \times 10 = 16450
\]
Therefore, the estimated value of the integral is 16,450.
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