To figure out the area under curves, we can use something called the Fundamental Theorem of Calculus (FTC). Here’s how to do it step by step:
Get to Know the FTC: The FTC tells us that if we have a continuous function (f) on the interval ([a, b]), we can find the area under (f) from (a) to (b) using this formula: [ A = \int_a^b f(x) , dx = F(b) - F(a) ] In this formula, (F) is an "antiderivative" of (f) (which is just a fancy way of saying it's a function that helps us find the area).
Let’s See an Example: Suppose we want to find the area under the curve of (f(x) = x^2) from (x = 1) to (x = 3).
First, we need to find an antiderivative: [ F(x) = \frac{x^3}{3} ]
Now, Calculate the Area: Next, we will use our formula to find the area: [ A = F(3) - F(1) ] Plugging in the numbers, we get: [ A = \left(\frac{3^3}{3}\right) - \left(\frac{1^3}{3}\right) ] This simplifies to: [ A = 9 - \frac{1}{3} = \frac{26}{3} ]
And that’s it! By following these steps, we can easily find the area under curves using calculus.
To figure out the area under curves, we can use something called the Fundamental Theorem of Calculus (FTC). Here’s how to do it step by step:
Get to Know the FTC: The FTC tells us that if we have a continuous function (f) on the interval ([a, b]), we can find the area under (f) from (a) to (b) using this formula: [ A = \int_a^b f(x) , dx = F(b) - F(a) ] In this formula, (F) is an "antiderivative" of (f) (which is just a fancy way of saying it's a function that helps us find the area).
Let’s See an Example: Suppose we want to find the area under the curve of (f(x) = x^2) from (x = 1) to (x = 3).
First, we need to find an antiderivative: [ F(x) = \frac{x^3}{3} ]
Now, Calculate the Area: Next, we will use our formula to find the area: [ A = F(3) - F(1) ] Plugging in the numbers, we get: [ A = \left(\frac{3^3}{3}\right) - \left(\frac{1^3}{3}\right) ] This simplifies to: [ A = 9 - \frac{1}{3} = \frac{26}{3} ]
And that’s it! By following these steps, we can easily find the area under curves using calculus.