How to Calculate Area Under a Curve Using Riemann Sums

TL;DR

To find the area under a curve with Riemann sums, split the interval into n rectangles of width Delta x = (b - a)/n, then add each rectangle's area f(x)·Delta x using left endpoints, right endpoints, or midpoints. For f(x) = x^2 + 1 on [0, 2] with 4 rectangles, Delta x = 0.5 and the left-endpoint estimate is 3.75, while the definite integral gives the exact area 14/3 (about 4.67). Read on to see how each endpoint choice changes the result.

Transcript

in this video we're going to focus on finding the area under the curve using riemann sums using left endpoints midpoints right in points sigma notation and limits and also by evaluating the definite integral so let's begin let's say if we have a function f of x is equal to let's say x squared plus one so this is basically a parabola that starts at ... Read More

Key Insights

• 🍹 The definite integral provides the exact value of the area under a curve, while Riemann sums offer approximations.
• 🗯️ The choice of endpoints (left, right, or midpoint) affects whether the approximation is an overestimation or underestimation of the actual area.
• 🍹 Increasing the number of rectangles in the Riemann sum improves the accuracy of the approximation.
• 🍹 The definite integral and Riemann sums are powerful tools that find applications in various mathematical and real-world scenarios.

Questions & Answers

Q: What is the difference between a left and right Riemann sum?

Both split the interval into rectangles of width Delta x, but they differ in which x-value sets each rectangle's height. On [0, 2] with 4 rectangles the left sum uses the points 0, 0.5, 1, and 1.5, while the right sum uses 0.5, 1.0, 1.5, and 2, dropping the leftmost point. For an increasing function the left endpoints give an under-approximation and the right endpoints give an over-approximation.

Q: How do you calculate a midpoint Riemann sum?

Use the midpoint of each subinterval as the rectangle's height instead of an endpoint. On [0, 2] with 4 rectangles of width 0.5, the four midpoints are 0.25, 0.75, 1.25, and 1.75. You then compute Delta x times the sum of f at those midpoints. The midpoint rule is usually more accurate than either the left or right endpoint rule.

Q: What is the formula for the width of each rectangle (Delta x)?

Delta x is the difference between the upper and lower limits divided by the number of rectangles: Delta x = (b - a)/n. In the example a = 0, b = 2, and n = 4, so Delta x = 2/4 = 0.5. This width is the same for every rectangle in the sum.

Q: How do you find the exact area under f(x) = x^2 + 1 from 0 to 2?

Evaluate the definite integral from 0 to 2 of x^2 + 1 dx. The antiderivative is x^3/3 + x, so you plug in 2 to get 8/3 + 2 and subtract the value at 0, which is 0. The result is 14/3, or about 4.67, which is the exact area.

Q: What does the sigma notation in a Riemann sum represent?

The sigma notation represents the sum of the areas of all the individual rectangles under the curve. Each rectangle's area is its height times its width, f(x) times Delta x, so the general form sums f(x_i)·Delta x from i = 1 to n. Adding these areas approximates the area between the curve and the x-axis.

Q: Why is the left-endpoint approximation an underestimate here?

For the increasing function f(x) = x^2 + 1, each left-endpoint rectangle sits below the curve, so its area falls short of the true region. Using 4 rectangles the left-endpoint sum comes out to 3.75, which is less than the exact answer of about 4.67. That gap shrinks as you increase the number of rectangles.

Q: How does the number of rectangles affect accuracy?

Increasing the number of rectangles n makes the approximation closer to the true area. With only 4 rectangles the left-endpoint estimate of 3.75 is not very close to 4.67, but raising n to 8 or more tightens the result. As n approaches infinity, the Riemann sum approaches the exact value given by the definite integral.

Summary & Key Takeaways

• The video demonstrates the process of finding the area under a curve using the definite integral from a to b of f(x)dx, where f(x) is the given function and a and b are the limits of integration.

• The concept of Riemann sums is introduced, where the area under the curve is approximated by summing the areas of multiple rectangles beneath the curve.

• The video explores different techniques for approximating the area using left endpoints, right endpoints, and midpoints of the intervals, highlighting their strengths and weaknesses.

• Examples are provided to illustrate the calculations and the resulting approximations of the area under the curve.