7. Let S = [0, 1] × [0, 1] and f : S → R be defined by f(x, y) = ( x + y, if x 2 ≤ y ≤ 2x 2 , 0, elsewhere. Show that f is integrable over S and calculate R S f(z)dz.
7. Let S = [0, 1] × [0, 1] and f : S → R be defined by f(x, y) = ( x + y, if x 2 ≤ y ≤ 2x 2 , 0, elsewhere. Show that f is integrable over S and calculate R S f(z)dz.
Let f : [0, 1] x [0, 1] + R be defined by f(x, y) = {1 if y = 23, 0 if y + x2 Show that f is integrable on (0, 1] x [0, 1]. You may take the previous problem as given
Let f : [0, 1] x [0, 1] → R be defined by f(x,y) - 1 if y=%, 0 if y#x2 Show that f is integrable on [0,1] [0,1]. You may take the previous problem as given
Consider f : [0, 1] x [0, 1] C R2 + R defined by f(x,y) = ſi if y is rational 2x if y is irrational Show that f is not integrable over R by the following steps: in (a) For each n > 1, find a Sn:= Eosi,jan f(a 6? b., in [0, 1] for 0 < i, j < n such that the Riemann sum converges as n + 0.[10 pts] n 1 n2 n i, ja (b)...
19.2. Let f : [a,b] → R be integrable. Show that rb 72 (r)dz, un 0O i=1 where a, b > 0 and h (b/a)1/n. In particular, calculate J-3r2dr by consid- ering a partition P which divides the interval [2, 3] into n parts in geometric progression at the points 2, 2h, 2h2,2h3,... ,2h"-1,2h" -3 19.2. Let f : [a,b] → R be integrable. Show that rb 72 (r)dz, un 0O i=1 where a, b > 0 and h (b/a)1/n....
1. Let f : [0, 1]2 → R be given by: 1 f(x,y) -»-< if x = 0 if x + y Show that f is integrable on [0, 1]2 and compute the value of the integral.
(c) Let F be the vector field on R given by F(x, y, z) = (2x +3y, z, 3y + z). (i) Calculate the divergence of F and the curl of F (ii) Let V be the region in IR enclosed by the plane I +2y +z S denote the closed surface that is the boundary of this region V. Sketch a picture of V and S. Then, using the Divergence Theorem, or otherwise, calculate 3 and the XY, YZ...
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to find .后f(x)dr Problem 2. Consider the function -2, zEQ 2, O f(r) = Show that f is not Riemann integrable on 0,1 but s Reemann integrable on this interval. Problem 3. (a) Let f be a real-valued function on a, b] such thatf()0 for all c, where c E [a, b Prove that f is Riemann integrable on a, b...
6. Let f:Q+R be integrable over the n-rectangle Q, and suppose f(x) > 0 for all x € Q. Show that ſo f > 0. (Be careful: it is possible for m (f) = 0 for a subrectangle RCQ, even when f >0.)
2. Let f(x,y) = e-r-u, 0 < x < oo, 0 < y < oo, zero elsewhere, be the pdf of X and Y. Then if Z = X + Y, compute (a) P(Z 0). (b) P(Z 6) (c) P(Z 2) (d) What is the pdf of Z?