## The area under the curve of a function can be found using the integral

### The area under the curve of f(x) = cos(x) on the interval [0, 2] can be found using the integral

The area under the curve of f(x) = cos(x) on the interval [0, 2] can be found using the integral:

int_0^2 f(x) dx = int_0^2 cos(x) dx

This can be evaluated using any of a variety of methods, such as numerical integration or by finding a closed-form solution.

### The area under the curve of f(x) = x^2 on the interval [0, 2] can be found using the integral

To find the area under the curve of f(x) = x^2 on the interval [0, 2], we can use the integral. The integral is a mathematical tool that allows us to find the area under a curve. In this case, we will be using the integral to find the area under the curve of f(x) = x^2 on the interval [0, 2].

To use the integral, we first need to identify the function that we are finding the area under. In this case, our function is f(x) = x^2. Next, we need to identify the interval over which we are finding the area. In this case, our interval is [0, 2].

Once we have identified our function and interval, we can use the integral to find the area under our curve. To do this, we need to take the integral of f(x) over our interval. In this case, our integral would be:

int f(x) dx = x^2 dx

The above equation is read as “the integral of f of x, with respect to x, equals x squared dx.” This equation tells us that to find the area under our curve (f(x)), we need to take the integral of x^2 over our interval [0, 2].

Once we have taken the integral of x^2 over our interval [0], we can then calculate the area under our curve. The final answer will depend on what units you are using for your calculations. In this case, if you are using square units (such as square centimeters), your answer will be in square units as well.

## The area under the curve of a function can also be approximated using the Riemann sum

The area under the curve of a function can be approximated using the Riemann sum. The Riemann sum is a method of approximating the area under the curve of a function by partitioning the interval into sub-intervals and summing the area of the rectangles formed by the function and the partition points.

### The area under the curve of f(x) = cos(x) on the interval [0, 2] can be approximated using the Riemann sum

The area under the curve of f(x) = cos(x) on the interval [0, 2] can be approximated using the Riemann sum. This sum is a series of rectangles with a width of Δx and a height of f(x). The total area is then the sum of the areas of all the rectangles.

### The area under the curve of f(x) = x^2 on the interval [0, 2] can be approximated using the Riemann sum

When approximating the area under a curve, mathematicians often use a technique called the Riemann sum. This involves partitioning the interval into smaller subintervals, and then calculating the area of each resulting rectangle. The total area is then approximated by adding up the areas of all the rectangles.

To calculate the area under the curve f(x) = x^2 on the interval [0, 2], we first need to partition the interval into smaller subintervals. For this example, we will use 4 subintervals, each of width 0.5. This gives us the following rectangles:

Rectangle 1: f(0) = 0, width = 0.5, height = 0.5^2 = 0.25

Rectangle 2: f(0.5) = 0.5^2 = 0.25, width = 0.5, height = 0.5^2 = 0.25

Rectangle 3: f(1) = 1^2 = 1, width = 0.5, height = 1^2 = 1

Rectangle 4: f(1.5) = 1.5^2 = 2.25, width = 0.5, height = 1.5^2 2.25

The total area is then approximated by adding up the areas of all rectangles:

Area ≈ (0 + 0 + 1 + 2) × (0 + 0 + 5 + 5) ÷ 4 ≈ 7 ÷ 4 ≈ 1 3/4