Table of Contents
- 1 Why is the integral equal to the area under a curve?
- 2 What is an integral area under a curve?
- 3 What does it mean if an integral is positive?
- 4 How is a definite integral related to area?
- 5 How do you find the area under a curve using integral?
- 6 Why can’t the area of an integral be negative?
- 7 What is the difference between an integral and a derivative?
Why is the integral equal to the area under a curve?
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”.
What is an integral area under a curve?
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.
What does it mean when you calculate the area under the curve and obtain a value that is negative?
When you are going backwards, your change in position is negative. Your total displacement is given by your total forwards movements minus your total backwards movements; these correspond to the regions above and below the t-axis. Thus signed area corresponds to displacement.
What does it mean if an integral is positive?
Integrals measure the area between the x-axis and the curve in question over a specified interval. If ALL of the area within the interval exists above the x-axis yet below the curve then the result is positive .
Definite integrals can be used to find the area under, over, or between curves. If a function is strictly positive, the area between it and the x axis is simply the definite integral. If it is simply negative, the area is -1 times the definite integral.
Is the integral of a positive function positive?
In the proof here a strictly positive function in (0,π) is integrated over this interval and the integral is claimed as a positive number. It seems intuitively obvious as the area enclosed by a continuous function’s graph lying entirely above the x-axis and the x-axis should not be zero.
How do you find the area under a curve using integral?
Note: Sometimes one is asked to find the total area bounded by a given curve. In that case, the definite integral could give you the result which is less than what is expected. For example- try calculating the area under the curve y = sin x from x = 0 to x = Π/2.
Why can’t the area of an integral be negative?
Simple answer – if the problem asks you to find the area, the negative values of the integrals make them positive . Area can’t be negative. If the problem is finding the value of the integral, the result is ok to be negative.
How to find the area of a curve with a negative number?
The curve y = f (x), completely below the x -axis. Shows a “typical” rectangle, Δx wide and y high. In this case, the integral gives a negative number. We need to take the absolute value of this to find our area: \\displaystyle {x}= {2} x = 2. The curve y = x 2 − 4, showing the portion under the curve from x = −1 to x = 2.
What is the difference between an integral and a derivative?
First: the integral is definedto be the (net signed) area under the curve. The definition in terms of Riemann sums is precisely designed to accomplish this. The integral is a limit, a number. There is, a priori, no connection whatsoever with derivatives.