Why is the area under a curve equal to the integral?

The Definite Integral A definite integral gives us the area between the x-axis a curve over a defined interval. is the width of the subintervals. It is important to keep in mind that the area under the curve can assume positive and negative values. It is more appropriate to call it “the net signed area”.

Is an indefinite integral the area under a curve?

Now remember that IF THE CURVE IS ABOVE the x-axes, the definite integral gives you the AREA under the curve. …

How do you prove that integral is the area under a curve?

The proof is beautiful and simple. Let A(t) be the area under the curve y=f(x) from some point, say 0, to t. Draw a picture of A(t) with A(t+d) for some small d>0. The derivative A'(t) is the limit of [A(t+d) – A(t)]/d as d->0.

Where is area under the curve?

The area under a curve between two points is found out by doing a definite integral between the two points. To find the area under the curve y = f(x) between x = a & x = b, integrate y = f(x) between the limits of a and b. This area can be calculated using integration with given limits.

What does area under the curve give?

Area below the axis: The area of the curve below the axis is a negative value and hence the modulus of the area is taken. The area of the curve y = f(x) below the x-axis and bounded by the x-axis is obtained by taking the limits a and b. The formula for the area above the curve and the x-axis is as follows.

Why is the area under a curve important?

Originally Answered: Why is it important to know the area of a curve in integral calculus? The area under a curve will indicate a number directly related to the data. Depending on the problem you are solving, it will be a solution to a question.

What does area under curve tell you?

A common use of the term “area under the curve” (AUC) is found in pharmacokinetic literature. It represents the area under the plasma concentration curve, also called the plasma concentration-time profile. The AUC is a measure of total systemic exposure to the drug.

What is area under the curve used for?

The AUC is a measure of total systemic exposure to the drug. AUC is one of several important pharmacokinetic terms that are used to describe and quantify aspects of the plasma concentration-time profile of an administered drug (and/or its metabolites, which may or may not be pharmacologically active themselves).

What is area under the curve physics?

when two curves coincide, the two objects have the same acceleration at that time. an object undergoing constant acceleration traces a horizontal line. zero slope implies motion with constant acceleration . the area under the curve equals the change in velocity .

What does area under a curve represent in calculus?

The area under the curve is defined as the region bounded by the function we’re working with, vertical lines representing the function’s bounds, and the -axis. The graph above shows the area under the curve of the continuous function, . The interval, , represents the vertical bounds of the function.

Why do we use definite integrals to find the area?

With definite integrals, we integrate a function between 2 points, and so we can find the precise value of the integral and there is no need for any unknown constant terms [the constant cancels out]. The area under a curve between two points can be found by doing a definite integral between the two points.

Can the area under a curve be negative?

This is not the correct answer for the area under the curve. It is the value of the integral, but clearly an area cannot be negative. It’s always best to sketch the curve before finding areas under curves.

Why does the area under a curve become the anti-derivative?

However when it comes to the area under a curve for some reason when you break it up into an infinite amount of rectangles, magically it turns into the anti-derivative. Can someone explain why that is the definition of the integral and how Newton figured this out?

How to find the area under a curve between two points?

The area under a curve between two points can be found by doing a definite integral between the two points. To find the area under the curve y = f(x) between x = a and x = b, integrate y = f(x) between the limits of a and b.