Integration by substitution
If the function u = g(x) has a continuous derivative on [a, b] and f is continuous on the range of g, then
Steps for Integrating by Substitution—Definite Integrals
1. Choose a substitution u = g(x), such as the inner part of a composite function.
2. Compute . Compute new u-limits of integration g(a) and g(b).
3. Re-write the integral in terms of u and du, with the u-limits of integration.
4. Find the resulting integral in terms of u.
5. Evaluate using the u-limits. No need to switch back to x’s!
Integration by Parts
Let
and
be two functions in
.
If
and
are continuous on
,
then
or
.
3. Applications of definite integrals
The areas of plane figures
If a continuous curve is defined in rectangular
coordinates by the equation
the area of the curvilinear trapezoid bounded by this curve, by two
vertical lines at the
points x=a and x =b and by a segment of the x-axis
,
is given by the formula
.
In the more general case, if the area S is bounded
by two continuous curves
and
and by two vertical lines x=a and x=b, where
when
,
we will then have:
.
If the curve is defined by equations in parametric
form
and
then the area of the curvilinear trapezoid bounded by this curve, by
two vertical lines (x=a and x=b), and by a segment of the x-axis is
expressed by the integral
,
where
and
are determined from the equations
and
on the interval
.
If a curve is defined in polar coordinates by the
equation
,
then the area of the sector AOB (Fig. 2), bounded by an arc of the
curve, and by two radius vectors OA and OB,
Fig. 2.
which correspond to the values
and
is expressed by the integral
.
The arc length of a curve
The arc length s of a curve y=f(x) contained between two points with abscissas x=a and x=b is
.
If a curve is represented by equations in parametric form and then the arc length s of the curve is
,
where and are values of the parameter that correspond to the extremities of the arc.
If a curve is defined by the equation in polar coordinates, then the arc length s is
,
where
and
are the values of the polar angle at the extreme points of the arc.
The volume of a solid of revolution
The volumes of solids formed by the revolution of a curvilinear trapezoid [bounded by the curve y=f(x), the x-axis and two vertical lines x=a and x=b] about the x-axis and y-axes are expressed, respectively, by the formulas:
and
.
The Area of a Surface of Revolution
The area of a surface formed by the rotation, about the x-axis, of an arc of the curve y=f(x) between the points x=a and x=b, is expressed by the formula
.
If a curve is represented by equations in
parametric form
then
,
where and are values of the parameter t.
Lecture 11 Differential equations of the first and second order. Homogeneous and non-homogeneous linear differential equations. Linear differential equations with constant coefficients