Let f(x) be a function defined on the interval [a, b] and F(x) be its anti-derivative. Then,

The above is called the second fundamental theorem of calculus.

is defined as the definite integral of f(x) from x = a to x = b. The numbers and b are called limits of integration. We write

**Evaluation of Definite Integrals by Substitution**

Consider a definite integral of the following form

**Step 1** Substitute g(x) = t

**⇒** g ‘(x) dx = dt

**Step 2** Find the limits of integration in new system of variable i.e.. the lower limit is g(a) and the upper limit is g(b) and the g(b) integral is now

**Step 3** Evaluate the integral, so obtained by usual method.

**properties of Definite Integral**

**13. Leibnitz Rule for Differentiation Under Integral Sign**

(a) If Φ(x) and ψ(x) are defined on [a, b] and differentiable for every x and f(t) is continuous, then

(b) If Φ(x) and ψ(x) are defined on [a, b] and differentiable for every x and f(t) is continuous, then

14. If f(x) ≥ 0 on the interval [a, b], then

15. If (x) ≤ Φ(x) for x ∈ [a, b], then

16. If at every point x of an interval [a, b] the inequalities

g(x) ≤ f(x) ≤ h(x)

are fulfilled, then

18. If m is the least value and M is the greatest value of the function f(x) on the interval [a, bl. (estimation of an integral), then

19. If f is continuous on [a, b], then there exists a number c in [a, b] at which

is called the mean value of the function f(x) on the interval [a, b].

20. If f^{2}2 (x) and g^{2} (x) are integrable on [a, b], then

21. Let a function f(x, α) be continuous for a ≤ x ≤ b and c ≤ α ≤ d.

Then, for any α ∈ [c, d], if

22. If f(t) is an odd function, then is an even function.

23. If f(t) is an even function, then is an odd function.

24. If f(t) is an even function, then for non-zero a, is not necessarily an odd function. It will be an odd function, if

25. If f(x) is continuous on [a, α], then is called an improper integral and is defined as

**27. Geometrically**, for f(x) > 0, the improper integral gives area of the figure bounded by the curve y = f(x), the axis and the straight line x = a.

**Integral Function**

Let f(x) be a continuous function defined on [a, b], then a function φ(x) defined by is called the integral function of the function f.

**Properties of Integral Function**

- The integral function of an integrable function is continuous.
- If φ(x) is the integral function of continuous function, then φ(x) is derivable and of φ ‘ = f(x) for all x ∈ [a, b].

**Gamma Function**

If n is a positive rational number, then the improper integral is defined as a gamma function and it is denoted by Γn

**Properties of Gamma Function**

**Summation of Series by Definite Integral**

The method to evaluate the integral, as limit of the sum of an infinite series is known as integration by first principle.

**Area of Bounded Region**

The space occupied by the curve along with the axis, under the given condition is called area of bounded region.

(i) The area bounded by the curve y = F(x) above the X-axis and between the lines x = a, x = b is given by

(ii) If the curve between the lines x = a, x = b lies below the X-axis, then the required area is given by

(iii) The area bounded by the curve x = F(y) right to the Y-axis and the lines y = c, y = d is given by

(iv) If the curve between the lines y = c, y = d left to the Y-axis, then the area is given by

(v) Area bounded by two curves y = F (x) and y = G (x) between x = a and x = b is given by

(vi) Area bounded by two curves x = f(y) and x = g(y) between y=c and y=d is given by

(vii) If F (x) ≥. G (x) in [a, c] and F (x) ≤ G (x) in [c,d], where a < c < b, then area of the region bounded by the curves is given as

**Area of Curves Given by Polar Equations**

Let f(θ) be a continuous function, θ ∈ (a, α), then the are t bounded by the curve r = f(θ) and radius α, β(α < β) is

**Area of Parametric Curves**

Let x = φ(t) and y = ψ(t) be two parametric curves, then area bounded by the curve, X-axis and ordinates x = φ(t_{1}), x = ψ(t_{2}) is

**Volume and Surface Area**

If We revolve any plane curve along any line, then solid so generated is called solid of revolution.

**1. Volume of Solid Revolution**

- The volume of the solid generated by revolution of the area bounded by the curve y = f(x), the axis of x and the ordinates it being given that f(x) is a continuous a function in the interval (a, b).
- The volume of the solid generated by revolution of the area bounded by the curve x = g(y), the axis of y and two abscissas y = c and y = d is it being given that g(y) is a continuous function in the interval (c, d).

**Surface of Solid Revolution**

(i) The surface of the solid generated by revolution of the area bounded by the curve y = f(x), the axis of x and the ordinates

is a continuous function in the interval (a, b).

(ii) The surface of the solid generated by revolution of the area bounded by the curve x = f (y), the axis of y and y = c, y = d is continuous function in the interval (c, d).

**Curve Sketching**

**1. symmetry**

- If powers of y in a equation of curve are all even, then curve is symmetrical about X-axis.
- If powers of x in a equation of curve are all even, then curve is symmetrical about Y-axis.
- When x is replaced by -x and y is replaced by -y, then curve is symmetrical in opposite quadrant.
- If x and y are interchanged and equation of curve remains unchanged curve is symmetrical about line y = x.

**2. Nature of Origin**

- If point (0, 0) satisfies the equation, then curve passes through origin.
- If curve passes through origin, then equate low st degree term to zero and get equation of tangent. If there are two tangents, then origin is a double point.

**3. Point of Intersection with Axes**

- Put y = 0 and get intersection with X-axis, put x = 0 and get intersection with Y-axis.
- Now, find equation of tangent at this point i. e. , shift origin to the point of intersection and equate the lowest degree term to zero.
- Find regions where curve does not exists. i. e., curve will not exit for those values of variable when makes the other imaginary or not defined.

**4. Asymptotes**

- Equate coefficient of highest power of x and get asymptote parallel to X-axis.
- Similarly equate coefficient of highest power of y and get asymptote parallel to Y-axis.

**5. The Sign of (dy/dx)**

Find points at which (dy/dx) vanishes or becomes infinite. It gives us the points where tangent is parallel or perpendicular to the X-axis.

**6. Points of Inflexion**

and solve the resulting equation.If some point of inflexion is there, then locate it exactly.

Taking in consideration of all above information, we draw an approximate shape of the curve.

**Shape of Some Curves is Given Below**

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