Given he toll aning graph at of (x)= x² -x*+3 on Hne の、nerual [い,3].agp3xinate [-1,3]. Angproximate (3 fex) dx by using the midp oint sums uhere shade e area that represents the mid paint Sums. b) Find Hre absolure error for using the midpoint juns (Recall *absulute error I True value - Approximahion I %3D
Arc Length
Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length.
Line Integral
A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.
Triple Integral
Examples:
![### Approximating the Integral of a Function Using Midpoint Sums
#### Problem Statement:
a) Given the following graph of \( f(x) = x^3 - 2x^2 + 3 \) on the interval \([-1, 3]\). Approximate \( \int_{-1}^{3} f(x) \, dx \) by using the midpoint sums where \( n = 4 \). Shade the area that represents the midpoint sums.
b) Find the absolute error for using the midpoint sums.
(Recall: Absolute error = | True value - Approximation |)
#### Explanation:
- **Midpoint Rule:**
- Divide the interval \([-1, 3]\) into 4 equal subintervals.
- Identify the midpoint of each subinterval.
- Calculate the area of rectangles using these midpoints and sum them up for an approximation of the integral.
- **Absolute Error:**
- Determine the true value of the integral.
- Subtract the approximated value from the true value.
- Take the absolute value of the result to find the absolute error.
This exercise helps in understanding numerical integration techniques and error analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2ce945b-ebf8-47d5-81d6-18edb03d27a8%2F3c2f930e-f930-4d48-abd3-185a5b44e079%2Fepgcle_processed.jpeg&w=3840&q=75)
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