Tag : integral calculus

Let g: R → R be a continuous function. Which one of the following is the solution of the differential equation d^2y/dx^2 + y = g(x) for x ∈ R, satisfying the conditions y(0) = 0, y'(0) = 1 ?
For a > b > 0, consider D = {(x,y,z) ∈ R^3 : x^2 + y^2 + z^2 ≤ a^2 and x^2 + y^2 ≥ b^2}. Then, the surface area of the boundary of the solid 𝐷 is
For a twice continuously differentiable function g: R → R, define u_(g)(x, y) = 1/y int_{-y}^{y}g(x + t)dt for (x, y) in R^2, y > 0. Which one of the following holds for all such g?
The area of the region R = {(x, y) ∈ R^2 : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 and 1/4 ≤ xy ≤ 1/2} is _____________ (rounded off to two decimal places).
Let S = {(x,y,z) ∈ R^3 : x^2 + y^2 + z^2 = 4, (x-1)^2 + y^2 ≤ 1, z ≥ 0}. Then, the surface area of 𝑆 equals _____________ (rounded off to two decimal places).
Let c > 0 be such that ∫^c_0{e^{s^2}}ds = 3 Then, the value of ∫^c_0(∫^c_x e^{x^2 + y^2}dy)dx equals _____________ (rounded off to two decimal places).