Inverse hyperbolic functions follow standard rules for integration. Remember, an inverse hyperbolic function can be written two ways. For example, inverse hyperbolic sine can be written as arcsinh or as sinh^(-1). Some people argue that the arcsinh form should be used because sinh^(-1) can be misinterpreted as 1/sinh. Whichever form you prefer, you see both, so you should be able to recognize both and understand that they mean the same thing.
Read MoreImproper integrals are just like definite integrals, except that the lower and/or upper limit of integration is infinite. Remember that a definite integral is an integral that we evaluate over a certain interval. An improper integral is just a definite integral where one end of the interval is +/-infinity.
Read MoreThink about the average value of a function as the average height the function attains above the x-axis. If the function were y=3, then the height of the function is always 3 everywhere, so the average height of the function would also be 3. When the function gets more complicated, we can use the average value formula to find its average height on [a,b].
Read MoreWe always lose the constant (term without a variable attached), when we take the derivative of a function. Which means we’re never going to get the constant back when we try to integrate our derivative. It’s lost forever. That is, unless we have an initial condition we can use to figure out what that constant was before we differentiated.
Read MoreQuadratic functions are functions in the form ax^2+bx+c=0. Integrating functions that include a quadratic can sometimes be a little difficult. There are three methods we’ll use to evaluate quadratic integrals: substitution, partial fractions, and trigonometric substitution. You should try using these techniques in the order listed above, because substitution is the easiest and fastest, and trigonometric substitution is the longest and most difficult.
Read MoreThe comparison theorem for improper integrals allows you to draw a conclusion about the convergence or divergence of an improper integral, without actually evaluating the integral itself. The trick is finding a comparison series that is either less than the original series and diverging, or greater than the original series and converging.
Read MoreConsumer and producer surplus are values that a company can calculate to see when they have excess demand or production. If a company can better balance demand and production, they can be more profitable. We’ll need to calculate the equilibrium quantity and equilibrium price before we can find consumer surplus and producer surplus.
Read MoreThe trapezoidal rule is one method we can use to approximate the area under a function over a given interval. If it’s difficult to find area exactly using an integral, we can use trapezoidal rule instead to estimate the integral. It’s called trapezoidal rule because we use trapezoids to estimate the area under the curve.
Read MoreWe can use integrals to find the surface area of the three-dimensional figure that’s created when we take a function and rotate it around an axis and over a certain interval. The formulas we use to find surface area of revolution are different depending on the form of the original function and the axis of rotation.
Read MoreU-substitution in definite integrals is just like substitution in indefinite integrals except that, since the variable is changed, the limits of integration must be changed as well. If you don’t change the limits of integration, then you’ll need to back-substitute for the original variable at the end.
Read MoreThe Theorem of Pappus tells us that the volume of a three-dimensional solid object that’s created by rotating a two-dimensional shape around an axis is given by V=Ad. V is the volume of the three-dimensional object, A is the area of the two-dimensional figure being revolved, and d is the distance traveled by the centroid of the two-dimensional figure.
Read MoreProbability density refers to the probability that a continuous random variable X will exist within a set of conditions. It follows that using the probability density equations will tell us the likelihood of an X existing in the interval [a,b].
Read MoreTo find the work required to stretch or compress an elastic spring, you’ll need to use Hooke’s Law. Every spring has its own spring constant k, and this spring constant is used in the Hooke’s Law formula.
Read MoreMost integrals need some work before you can even begin the integration. They have to be transformed or manipulated in order to reduce the function’s form into some simpler form. U-substitution is the simplest tool we have to transform integrals.
Read MoreIn this video we talk about why the integration by parts formula actually works.
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