Calculus Answers

Evaluate the following integral after converting to polar coordinates: ∫(0 to 1)∫(y to sqrt(2-y^2)) (x+y) dx dy

Sketch the region of integration, carefully with appropriate labels, and then perform the following integral: ∫(0 to pi/4)∫(0 to 4secO) r dr dO

Evaluate ∫∫Rcos(x^2+y^2)dA, where R is the region above the x-axis within the circle x^2+y^2= 25.

f(x,y)=−2x^2+4y^2
find the value of the directional derivative at the point (3,4) in the direction given by the angle θ=2π/5. More specifically, find the directional derivative of f at the point (3,4) in the direction of the unit vector determined by the angle θ in polar coordinates.

Find the partial derivative of f(x,y,z)=x^2y+6z^3xy

Evaluate the integral using a rational function ∫▒(cos x)/(√sin+ ∛(sin⁡x )) dx

(a) Create a function that is a product of two non-constant functions that would NOT require the product rule to differentiate it.

(b) Differentiate your function from part (a) without using the product rule.

(c) Differentiate your function from part (a) by using the product rule and confirm your result is equivalent to your result in part (b) by simplifying.

(a) Create a function that is a product of two non-constant functions that would NOT require the product rule to differentiate it.

(b) Differentiate your function from part (a) without using the product rule.

(c) Differentiate your function from part (a) by using the product rule and confirm your result is equivalent to your result in part (b) by simplifying.