Match real-domain inverse hyperbolic integral formulas, and work through inverse sech, inverse csch, and inverse tanh examples.
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Set up improper integrals with infinite bounds or unbounded integrands, and determine whether each one converges.
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Divide the definite integral by the interval length, see why the formula works, and evaluate a worked example.
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Find a family of antiderivatives, use the initial condition to solve for the constant, and check the particular solution.
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Choose substitution, partial fractions, or trig substitution when a quadratic is in a denominator or under a square root.
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Compare nonnegative functions to prove convergence or divergence. Work through power and exponential bounds and the p = 1 case.
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Find equilibrium from the demand and supply curves, then integrate the two areas that give consumer and producer surplus.
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Find the subinterval width, weight interior heights twice, and use four trapezoids to approximate a definite integral.
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Set up the surface-area integral using the distance to the axis and the arc-length factor. Work through y = x³ rotated about the x-axis.
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Use u-substitution to evaluate a definite integral. Work through changing the bounds and back-substituting before applying the original bounds.
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Use Pappus’s centroid theorem to find the volume of a cone. Find the generating triangle’s centroid by integration, then multiply its area by the distance traveled.
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Check that a continuous density is nonnegative and has total area 1. Then integrate over a specific interval to find its probability.
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Use Hooke’s law to find the spring constant and integrate force over displacement. Work through one stretch and one compression example.
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Learn how to choose u, match du, integrate, and substitute back with power, rational, exponential, and trigonometric examples.
Read MoreSee how a triple integral adds up contributions over a solid. Set bounds for a tetrahedron and learn when the result gives volume or mass.
Read MoreSee how a double integral adds up volume columns over a region. Learn when it gives volume, when it gives signed accumulation, and how to set the bounds.
Read MoreLearn what dx means in integral notation, how it identifies the variable of integration, and how it connects to the widths in a Riemann sum.
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