Integral Help

The graph above shows the base of an object. Compute the exact value of the volume of the object, given that cross sections (perpendicular to the base) are squares. #CrossSection #Area #Volume #Integrals

A ball of radius 13 has a round hole of radius 9 drilled through its center. #Radius #Calculus #Integral #Volume #Integrations

Calculus Help: If I_n=∫ x^n e^x dx ,n∈NProve that I_n=x^n e^x-nI_(n-1) #Parts #Integrals #Techniques

#DoubleIntegrals #Integrations #Reversed #Calculus

An object of mass m is dropped into the sea at the surface. It is subject to a constant gravitational acceleration g and water resistance. (a) (4 points) If the water resistance is proportional to the object's velocity v(t) with coefficient b ≫ 0, so that Newton's second law gives b dv/dt =g- b/m v show that the object's velocity as a function of time t is given by v(t) = mg/b (1-e^(-bt/m)). We take downward to be the positive direction and the object is at rest when it is dropped at t=0. (b) (4 points) If instead the water resistance is quadratic in the velocity, so that Newton's second law gives dv/dt =g- b/m v^2 dt m dv dt find the object's velocity as a function of time t, again assuming that the object is dropped from rest. [Hint: Write Newtons' law as = g(1 − k^2v^2) where k = √b/mg. Then the algebra is a little less messy. After obtaining the final answer in terms of k, remember to replace k by √b/mg.] (c) (2 points) If the object could fall forever under these conditions without hitting the bottom, show that as t→ ∞ the velocity in part (a) approaches terminal velocity v → mg/b. What terminal velocity does the velocity in part (b) approach? #CalculusHelp #Integrals #differentialequations #PartialFractions

Here is the technique to solve these 4 integrals by integration by substitution ∫ cosx cos^2⁡ (sinx) dx ∫ sinx cos^2⁡ (cosx) dx ∫ cosx sin^2⁡ (sinx) dx ∫ sinx sin^2⁡ (cosx) dx #CalculusHelp #Integral #Integrations #IntegrationbySubstitution #Substitution

Calculus Help: Integral ∫ sin2x/e^x dx - Integration by parts - Techniques #Integral #Integrations #Calculus #Formula #Parts

Calculus Help: Integral ∫ 1/(3+2tanx) dx - Find the coefficient - Integration by substitution #Integral #Integrations #Calculus #Antiderivative

Calculus Help: Integral ∫ (x+1) e^2x e^(x^2 ) dx - Integration by substitution #Calculus #Integral #Integrations #Substitution

∫▒〖∛(e^arccot⁡(4x) )/(16x^2+1) dx〗 Integration by substitution #Integration #Calculus #Substitution #Techniques