What do you use integration for

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.

Additional exercises 9 Applications of Integration 1. Area between curves 2. Distance, Velocity, Acceleration 3. Volume 4. Average value of a function 5. Work 6. Center of Mass 7. Kinetic energy; improper integrals 8. Probability 9. Arc Length Polar Coordinates 2.

Slopes in polar coordinates 3. Areas in polar coordinates 4. Parametric Equations 5. Calculus with Parametric Equations 11 Sequences and Series 1. Sequences 2. Integration can be used to find areas, volumes, central points and many useful things. But it is easiest to start with finding the area between a function and the x-axis like this:. What is the area? So you should really know about Derivatives before reading more!

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 :.

The arc length is first approximated using line segments, which generates a Riemann sum. Taking a limit then gives us the definite integral formula. The same process can be applied to functions of y. The concepts used to calculate the arc length can be generalized to find the surface area of a surface of revolution.

The integrals generated by both the arc length and surface area formulas are often difficult to evaluate. Several physical applications of the definite integral are common in engineering and physics.

Definite integrals can be used to determine the mass of an object if its density function is known. Work can also be calculated from integrating a force function, or when counteracting the force of gravity, as in a pumping problem. Definite integrals can also be used to calculate the force exerted on an object submerged in a liquid. The basic idea of the center of mass is the notion of a balancing point.