## What is the point of fundamental theorem of calculus?

As mentioned earlier, the Fundamental Theorem of Calculus is an extremely powerful theorem that establishes the relationship between differentiation and integration, and gives us a way to evaluate definite integrals without using Riemann sums or calculating areas.

**What is the difference between Riemann sums and definite integral?**

Definite integrals represent the exact area under a given curve, and Riemann sums are used to approximate those areas.

### What is Riemann sum in calculus?

A Riemann sum is an approximation of a region’s area, obtained by adding up the areas of multiple simplified slices of the region. It is applied in calculus to formalize the method of exhaustion, used to determine the area of a region. This process yields the integral, which computes the value of the area exactly.

**How many fundamental theorems of calculus are there?**

The four fundamental theorems of vector calculus are generalizations of the fundamental theorem of calculus.

#### What does the first fundamental theorem of calculus tell us?

The first fundamental theorem says that the integral of the derivative is the function; or, more precisely, that it’s the difference between two outputs of that function. / b F = f, then f(x) dx = F (b) – F (a).

**Why are there two parts to the fundamental theorem of calculus?**

That’s what the two parts are: loosely stated, The first part shows that differentiating an integral gives the original function. The second part shows that integrating a derivative gives the original function.

## How do you know if a Riemann sum is an overestimate or underestimate?

If the graph is increasing on the interval, then the left-sum is an underestimate of the actual value and the right-sum is an overestimate. If the curve is decreasing then the right-sums are underestimates and the left-sums are overestimates.

**How do you evaluate an integral using Riemann sums?**

59 second clip suggested6:16Calculating a Definite Integral Using Riemann Sums – Part 1 – YouTubeYouTube

### How do you calculate a Riemann sum?

Riemann Sums Using Rules (Left – Right – Midpoint).

- When the n subintervals have equal length, Δxi=Δx=b−an.
- The i th term of the partition is xi=a+(i−1)Δx.
- The Left Hand Rule summation is: n∑i=1f(xi)Δx.
- The Right Hand Rule summation is: n∑i=1f(xi+1)Δx.
- The Midpoint Rule summation is: n∑i=1f(xi+xi+12)Δx.

**What is the difference between first and second fundamental theorem of calculus?**

There are two parts to the theorem. The first part deals with the derivative of an antiderivative, while the second part deals with the relationship between antiderivatives and definite integrals.