Why integration is used

Shell integration also called the shell method is a means of calculating the volume of a solid of revolution when integrating perpendicular to the axis of revolution.

When integrating parallel to the axis of revolution, you should use the disk method. While less intuitive than disk integration, it usually produces simpler integrals. Intuitively speaking, part of the graph of a function is rotated around an axis, and is modeled by an infinite number of cylindrical shells, all infinitely thin.

The Shell Method : Calculating volume using the shell method. By adding the volumes of all these infinitely thin cylinders, we can calculate the volume of the solid formed by the revolution.

Forces may do work on a system. This scalar product of force and velocity is classified as instantaneous power delivered by the force. Just as velocities may be integrated over time to obtain a total distance, by the fundamental theorem of calculus, the total work along a path is similarly the time-integral of instantaneous power applied along the trajectory of the point of application.

Work is the result of a force on a point that moves through a distance. The sum of these small amounts of work over the trajectory of the point yields the work:. This integral is computed along the trajectory of the particle, and is therefore said to be path-dependent. This calculation can be generalized for a constant force that is not directed along the line, followed by the particle. This is to say:. Work done by the restoring force leads to increase in the kinetic energy of the object.

The Chain Rule 4 Transcendental Functions 1. Trigonometric Functions 2. A hard limit 4. Derivatives of the Trigonometric Functions 6. Exponential and Logarithmic functions 7. Derivatives of the exponential and logarithmic functions 8. Implicit Differentiation 9. Inverse Trigonometric Functions Limits revisited Hyperbolic Functions 5 Curve Sketching 1. Maxima and Minima 2. The first derivative test 3. The second derivative test 4.

Concavity and inflection points 5. Optimization 2. Related Rates 3. Newton's Method 4. Linear Approximations 5. The Mean Value Theorem 7 Integration 1. Two examples 2. The Fundamental Theorem of Calculus 3.

Some Properties of Integrals 8 Techniques of Integration 1. Substitution 2. Powers of sine and cosine 3. Trigonometric Substitutions 4. Integration by Parts 5. Rational Functions 6. Numerical Integration 7. Laplace Transform of an integral is an important process in electronics.

We begin with a discussion of the differential, because it involves some of the concepts and notation used in the study of integration. The Differential ». Name optional. The Differential 2. Antiderivatives and The Indefinite Integral 3. The Area Under a Curve 4. The Definite Integral 5. Trapezoidal Rule 6. Riemann Sums 6b. Fundamental Theorem of Calculus Applet 7. After the Integral Symbol we put the function we want to find the integral of called the Integrand ,.

It is the "Constant of Integration". It is there because of all the functions whose derivative is 2x :. So when we reverse the operation to find the integral we only know 2x , but there could have been a constant of any value.

We can integrate that flow add up all the little bits of water to give us the volume of water in the tank.