A function f is defined over an interval [a, b]. (a) Graph...

A function $f$ is defined over an interval $[a, b]$. (a) Graph $f$, indicating the area $A$ under $f$ from $a$ to $b$. (b) Approximate the area $A$ by partitioning $[a, b]$ into four subintervals of equal length and choosing $u$ as the left endpoint of each subinterval. (c) Approximate the area $A$ by partitioning $[a, b]$ into eight subintervals of equal length and choosing $u$ as the left endpoint of each subinterval. (d) Express the area $A$ as an integral. (e) Use a graphing utility to approximate the integral. $f(x)=\cos x,\left[0, \frac{\pi}{2}\right]$

A function $f$ is defined over an interval $[a, b]$.
(a) Graph $f$, indicating the area $A$ under $f$ from $a$ to $b$.
(b) Approximate the area $A$ by partitioning $[a, b]$ into four subintervals of equal length and choosing $u$ as the left endpoint of each subinterval.
(c) Approximate the area $A$ by partitioning $[a, b]$ into eight subintervals of equal length and choosing $u$ as the left endpoint of each subinterval.
(d) Express the area $A$ as an integral.
(e) Use a graphing utility to approximate the integral.
$f(x)=\cos x,\left[0, \frac{\pi}{2}\right]$

Key Concepts

Definite Integral

The definite integral of a function over an interval is a fundamental concept in calculus that represents the net area under the curve of that function between two given points. It is defined as the limit of Riemann sums as the number of subintervals approaches infinity, thereby providing an exact measure of accumulated quantity, such as area, distance, or other related measures.

Riemann Sum

A Riemann sum is an approximation method for calculating the value of a definite integral by dividing the interval into subintervals, choosing sample points within these subintervals, and summing the products of the function values at these points and the widths of the subintervals. It is a foundational concept in understanding how integrals are formally defined and computed.

Partitioning the Interval

Partitioning an interval involves dividing a larger interval into smaller subintervals of equal or varying lengths. This technique is crucial in numerical integration because it lays the groundwork for approximating areas under curves, whether using Riemann sums, trapezoidal rules, or other numerical methods.

Left Endpoint Approximation

Left endpoint approximation is a specific strategy for constructing a Riemann sum where the sample point in each subinterval is chosen to be the left endpoint. This method is often used in introductory calculus problems to approximate the area under a curve, though it may either overestimate or underestimate the actual integral depending on the function's behavior.

Graphical Interpretation of Integrals

Graphing the function along with the area under the curve helps in visualizing the concept of integration. By shading the region between the function and the horizontal axis over the defined interval, one can better understand the geometric interpretation of integrals as areas, as well as the effects of various approximation methods.

Using Graphing Utilities for Approximation

Graphing utilities and computational tools are widely used to approximate the values of definite integrals, especially when analytical solutions are difficult or impossible to find. These tools use numerical methods to compute the sums and visualize the function and its corresponding area, making them essential in both educational and practical applications of calculus.

Transcript

00:01 So our function is sine of x, and we're going from 0 to pi.

00:08 And so we know that we're going to want to have for x and for f of x, we want to have zero, and we want pi over 4, and we want pi over 2, and we want 3 pi over 4, and we want pi, when we graph those, and the sign of that is zero.

00:30 This is square root of 2 over 2.

00:32 This is 1.

00:34 This is also square root of 2 over 2, and this is 0.

00:38 And so when we look at the graph of that function, and here's pi, we're getting one portion of that wave.

00:49 So 0, and this is about 0 .7, and this is up to 1, and this is about 0 .7, and that's back down to 0.

00:57 So we're getting half of a wave, and we're finding all this area.

01:05 Now, in part b, we're finding four different subdivisions that are equal.

01:14 Each one is pi over four wide, and we're using our left endpoint to describe what the area is.

01:24 So we're finding the area of those four rectangles, and that is zero.

01:31 Tall, one unit wide.

01:35 This is square root of two over two units tall, one unit wide.

01:40 This is one unit tall, one unit wide.