In laymen's terms, what is an integral?

Question

"anti-derivative" isn't helping. Hypothetical Examples would help.

Answer 1

It's the area under the curve.

The derivative of a function gives the slope of the function at each point.

The integral of a function gives the area under the function's graph (down to the x-axis) between any two given x-values.

Consider the equation y=3, a horizontal line. The antiderivative is 3x+c. The integral between the points 0 and 10 is: (3*10+c) - (3*0+c) = 30. The rectangle defined by that line and the X axis, from 0 to 10, has height 3 and length 10, has area 30.

It's really easy to compute the area from a line without using an integral, but when the function is not simply linear, calculating the area under it becomes impossible without that technique.

Answer 2

There are two separate things the word "integral" can be used to mean:

A "definite integral," when evaluated, yields a number. These integrals are the ones with numbers on the top and bottom of the integral sign.

An "indefinite integral" is a function. These integrals are the ones without numbers at the top and bottom of the integral sign.

I'll talk about definite integrals first.

Interpretation 1 (Geometric): The definite integral from a to b of a function f, provided f is always nonnegative, is the area of the region under the graph of f, above the x-axis, and between the lines x = a and x = b.

To take a simple example, if we have f(x) = 2, then the definite integral from 0 to 3 of f will be 6. (The area under the line y = 2, over the line y = 0, and between the lines x = 0 and x = 3 forms a rectangle of area 6.)

Interpretation 2 (Measuring change): If f measures how fast something is changing, then the definite integral from a to b of f measures how much that something changed as x went from a to b.

For example, if v(t) measures how fast a car was going t hours after noon, then the definite integral from 0 to 3 of v(t) will be how far that car traveled between noon and 3:00 PM.

To make this example more concrete, suppose that a car starts moving at noon, at a constant rate of 30 miles per hour; then we have v(t) = 30 (because the car is moving at 30 miles per hour at any time t). The definite integral from 0 to 3 of v(t) will, then, be 90 (because from noon to 3:00, the car will have traveled 90 miles).


A note about indefinite integrals: The indefinite integral of f is a function which, when you input x, it will output some particular definite integral of f. (This is very much like the behavior of the derivative as a function; the derivative is a function which, when you input x, outputs the rate of change of f at that particular point x.)

It is a small miracle that indefinite integrals and antiderivatives are (roughly speaking) the same thing. That is, one (very convenient) way of computing a definite integral of f is to first find an antiderivative of f, and then evaluate that antiderivative at the two points you have on your definite integral.

For example, when we had the car traveling at 30 miles per hour, we could instead have solved the problem as follows:

-We know v(t) = 30
-An antiderivative of v(t) is s(t) = 30t, because the derivative of s(t) is 30
-We are interested in the distance traveled between noon and 3 P.M. (That is, t = 0 to t = 3), so we compute s(3) - s(0). This is 90, so the car traveled 90 miles.

This example seems somewhat contrived given that the problem is so simple in this case, but one can also use integrals to compute how far the car traveled when the function describing its velocity is extremely complex, such as v(t) = 8t^7 + 4t^2 + 2^t. (In this case, it is virtually impossible to find out how far the car traveled just by reasoning--one must instead compute an antiderivative and evaluate it at the appropriate endpoints.)

Answer 3

When speaking of the derivative we speak of an instantaneous rate. Instantaneous is the concept that the interval of time is narrowed to an infinitesimally small amount. Now when working with area the sum of an area in an irregular shape can be found. The irregular shape is filled with rectangles and each rectangle has its own amount of area. As the number of rectangles increases to an infinitely large number and the closer the approximate area is calculated. This is the opposite of the derivative so we use the antiderivative.
The Area under the curve ∫1/0 x dx. ( the area under the curve between 1 and zero.) I can't use superscript or subscript. So pretend the 1 is at the top of the symbol and the zero is at the bottom. The antiderivative of X is 2X^2 so we use it :
2(1)^2 - 2( 0 )^2 = 1/2 and the area is 1/2 units.

Answer 4

yes it is an anti derivative for example by the power rule
dy/dx(x^5) = 5x^4 the integral is just the inverse operation of this because it gives x^5 back when you perform the integration
process on the expression on the derivative.

Answer 5

An integral represents the area under a curve divided into small rectangles denoted as dx.

Answer 6

It is area under the curve representing the integrand, the vertical lines at the ends of the interval and the x-axiswhich represents the independent variable..

Answer 7

Is the area under the curve, which you can calculate it depending the shape of that curve...