Calculus Answers

If P is the function defined by P(x)=a1+2a2x⋅⋅⋅ +(n+1)an+1xn write down an expression for \int^b_aP(x)dx Justify your answer

If f is continuous except at finitely many points in I,then f is Riemann-integrable.

Any continuous function f defined on a closed interval I=[a,b] is uniformly continuous.Prove

The function f(x)=x is uniformly continuous on R but the function f(x)=x^2 is not.Prove

If f is continuous on [a,b]t hen \underline_{\int_a^b}f(x)dx=\overline{\int_a^b}f(x)dx

Prove that Any two partitions have a common refinement.

Find the average value of f(x,y)=2 x^4 y^5 over the rectangle R with vertices (−6,0),(−6,1),(6,0),(6,1).
Average value =

Electric charge is distributed over the disk
x^2+y^2≤2 so that the charge density at (x,y) is σ(x,y)=17+x^2+y^2 coulombs per square meter.
Find the total charge on the disk.

Using polar coordinates, evaluate the integral ∫∫R sin(x^2+y^2)dA where R is the region 4≤x^2+y^2≤36.

For the following regions R and surface densities σ, find the total mass.
(a)R: 0≤y≤sin(pi x/L); 0≤x≤L; σ(x,y) =y.
(b) The region between the semicircles y=sqrt(1-x^2) and y=sqrt(4-x^2) and the segments of the x-axis joining them. The surface density is equal to the distance from any point to the origin.