Limits Using Definite Integrals

Finding the integral involves the sum of the areas. Similarly, definite integrals are helpful to calculate the area within the start and endpoints between which the area is occupied; this can be said to be the limits. If we take [a, b], as the limit points to find the area of the curve f(x), concerning the x-axis, the expression of definite integrals becomes ∫baf(x)dx∫abf(x)dx. Let’s go into detail about definite integrals and how to find the limits using definite integrals.

Definition of Definite Integrals

In the ninth standard, we used to calculate the area under the curve between the start and endpoints by dividing the area of the rectangles and then summing them up. In other words, we used to add the area divided by us into a limitless number of rectangles. The accuracy of the area was directly proportional to the number of rectangles. Thus, we can define definite integrals as the area between the two limits in the curve.

Finding the Definite Integral

To find the definite integral, we use one of two formulas for definite integrals, as follows:

• Fundamental theorem of calculus 

∫abf (x)dx=F(b)−F(a), where F'(x) = f(x)

• Definite integral as a limit sum which can also be called the limit using definite integrals  

abf (x)dx=n∞ r=1nf(a+rh)/h

where h=b-an

Definite Integrals with the Fundamental Theorem of Calculus Formula 

The fundamental theorem of calculus is the easiest method to calculate definite integrals. Using it, we have to first find the antiderivative of f(x) (and represent it F(x)), substitute the upper limit first, and then by individually by lowering the limits, the results in the order must be subtracted.

We calculate the A definite integrals ∫abf (x)dx by using the fundamental theorem of calculus (FTC). This formula says ∫abf (x)dx=F(b)−F(a), where F'(x) = f(x)

Example: Solve ∫01 x2dx using this definite integrals formula.

As per the rules of solving definite integrals using the fundamental theorem of calculus,

First, let’s solve the ∫x2dx using the integral formulas. 

After solving, we get, ∫x2dx = x3/3 + C.

Now, we will substitute the upper and lower limits to find the difference.

∫01 x2dx= (13/3 + C) – (03/3 + C) = 1/3.

C is the constant integral C which is always cancelled while solving the definite integral, so we can ignore it. 

Definite Integral as a Limit Sum Formula

Definite integrals are used to find the area of the curve, which has two limits. The area is measured by calculating the enclosed number of rectangles in the two limits. Using this concept, to evaluate definite integrals ∫abf(x)dx, the curve area is divided into many rectangles by dividing [a, b] into an infinite number of subintervals. Thus, the definite integral as a limit sum formula is:

∫abf (x)dx=limn→∞ ∑r=1n1hf(a+rh) 

Here h=b−anh=b−an is the length of each subinterval.

Example: We will evaluate 01x2dx using the above formula.

Comparing the integrals with abf (x)dx, [a, b] = [0, 1] and f(x) = x2. Then h = (1 – 0)/n = 1/n. Applying the above formula,

01x2dx=n∞ r=1nf(0+r/n) /n

= n∞ r=1n(r/n)2 /n

=n∞1/n3r=1nr2 

= n∞1/n3⋅n(n+1)(2n+1)/6 (using summation formulas)

=n∞(1/n3)n3(1+1/n)(2+1/n)/6 

= (1+0)(2+0)/6

= 1/3

Rules of Definite Integrals

To calculate definite integrals, we must also know its rules as they help determine the integrals for properties and find the integral sum of functions, a function multiplied by a constant, and for even and odd functions. The following rules are helpful to calculate the definite integral:

• abf (x).dx=abf (t).dt
• ab-f (x).dx=−abf (x).dx
• abc f (x).dx=cabf (x).dx
• abf (x)±g(x).dx=abf (x).dx±abg(x).dx
• abf (x).dx=∫acf (x).dx+∫cbf (x).dx
• abf (x).dx=abf (a+b−x).dx
•  0a f(x).dx= 0a f (a−x).dx (This is a formula derived from the above formula.)
•  02a f (x).dx=2 0af(x).dx  if f(2a – x) = f(x)
• 02a f (x).dx=0 if f(2a – x) = -f(x).
• -aaf(x).dx=2 ∫0a f(x).dx, if f(x) is an even function (i.e., f(-x) = f(x)).
• -aa f(x).dx=0, if f(x) is an odd function (i.e., f(-x) = -f(x)).

Conclusion

Thus, by using definite integrals, we can calculate the curve area under the limits. From the example of a limit using definite integrals, we also understand how to evaluate definite integrals to calculate the limits.