This chapter, Application of Integrals, for CBSE Class 12 Mathematics, delves into using definite integrals to solve real-world problems. We'll explore how to calculate areas of various bounded regions, including those enclosed by two curves, and even those defined by polar and parametric equations. The notes also cover finding the volume and surface area of solids generated when a curve is revolved around an axis. A key aspect is understanding curve sketching, which involves analyzing symmetry, intercepts, asymptotes, and the behavior of the function to visualize these shapes accurately. This chapter builds upon the fundamental theorem of calculus and properties of definite integrals, providing a practical application of integration techniques essential for exam success.
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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
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.
(a) If ?(x) and ?(x) are defined on [a, b] and differentiable for every x and f(t) is continuous,
then
(b) If ?(x) and ?(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) ? ?(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 f22 (x) and g2 (x) are integrable on [a, b], then
21. Let a function f(x, ?) be continuous for a ? x ? b and c ? ? ? d.
Then, for any ? ? [c, d], if
22. If f(t) is an odd function, thenis an even function.
23. If f(t) is an even function, thenis 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, ?], thenis called an improper integral and is defined as 27. Geometrically, for f(x) > 0, the improper integralgives area of the figure
bounded by the curve y = f(x), the axis and the straight line x = a.
Let f(x) be a continuous function defined on [a, b], then a function ?(x) defined
by
is called the integral function of the function f.
1. The integral function of an integrable function is continuous. 2. If ?(x) is the integral function of continuous function, then ?(x) is derivable and of ? ‘ = f(x) for all x ? [a, b].
If n is a positive rational number, then the improper integral
is defined as a
gamma function and it is denoted by ?n 
The method to evaluate the integral, as limit of the sum of an infinite series is known as
integration by first principle.
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
Let f(?) be a continuous function, ? ? (a, ?), then the are t bounded by the curve r = f(?) and
<?) is
Let x = ?(t) and y = ?(t) be two parametric curves, then area bounded by the curve, X-axis and
ordinates x = ?(t1), x = ?(t2) is
If We revolve any plane curve along any line, then solid so generated is called solid of revolution.
1. 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). 2. 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).
(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 ordinatesis 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 iscontinuous function in the interval
(c, d).
1. If powers of y in a equation of curve are all even, then curve is symmetrical about Xaxis. 2. If powers of x in a equation of curve are all even, then curve is symmetrical about Yaxis. 3. When x is replaced by -x and y is replaced by -y, then curve is symmetrical in opposite quadrant. 4. If x and y are interchanged and equation of curve remains unchanged curve is symmetrical about line y = x.
1. If point (0, 0) satisfies the equation, then curve passes through origin. 2. 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.
1. Put y = 0 and get intersection with X-axis, put x = 0 and get intersection with Y-axis. 2. 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. 3. 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.
1. Equate coefficient of highest power of x and get asymptote parallel to X-axis. 2. Similarly equate coefficient of highest power of y and get asymptote parallel to Y-axis.
Find points at which (dy/dx) vanishes or becomes infinite. It gives us the points where tngent is parallel or perpendicular to the X-axis.
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
CBSE Class 12 Maths Chapter 8 covers the Application of Integrals, focusing on calculating areas, volumes, and understanding curve sketching.
Evaluation by substitution involves replacing a part of the integrand with a new variable 't', finding the differential 'dt', changing the limits of integration to correspond to 't', and then evaluating the new integral.
The notes list several properties, including Leibnitz Rule for differentiation under the integral sign, properties related to the sign of the function, and estimation of integrals.
An integral function, denoted by ?(x), is defined as the definite integral of a continuous function f(x) from a constant 'a' to 'x', and it is itself continuous and derivable.
The area can be calculated using definite integrals, depending on whether the curve is above or below the x-axis, or bounded by two curves, or described by polar or parametric equations.
Curve sketching involves analyzing symmetry, behavior at the origin, intersections with axes, asymptotes, the sign of dy/dx, and points of inflection to draw an approximate shape of a curve.
The Gamma function is an improper integral defined for positive rational numbers, denoted by ?n, and it has several important properties.
