12. Evaluate the integral.
∫(sec ²(t) i+t(t²+1)⁵ j+t⁵ ln (t) k) d t
13.
Find r(t) if r'(t)=t⁵ i+e^{t} j+3te³t k and r(0)=i+j+k.
14.
Find f'(2), where f(t)=u(t) · v(t), u(2)=(1,2,-1), u'(2)=(3,1,7), and v(t)=(t, t2), t3).
Evaluate the integral.∫(sec ²(t) i+t(t²+1)⁵ j+t⁴ ln (t) k) d t
Evaluate the integral.∫(sec²(t) i+t(t²+1)⁵j+t⁴ln(t)k)dt
Let u(t) =t^3 i + ln(t) j + e^2t k and v(t) = 1/t^3 i + 2 j + t k 2. Let u(t) - ti+In(t)j+ et k and Compute the derivative of the dot product f u(t)v() in two ways and confirm they agree: Compute the dot product u(t) v(t) first and then differentiate the result. . Alternatively, use the following "Dot Product Rule" u(t) v(t)] u'(t) v(t)+ u(t) v'(t) Aside: It's worth noting that there are other forms...
Evaluate the line integral ∫C.F·dr, where C is given by the vector function r(t).F(x, y, z) = sin(x) i + cos(y) j + xz k r(t) = t3 i- t3j + tk, 0 ≤ t ≤ 1 .
Part A Evaluate the integral: I= j (+5 + 5) (8 (t) + 128 (t – 1)) dt. Express your answer using three significant figures. View Available Hint(s) VO AXO Ut vec 2 ? Submit Part B Evaluate the integral: I = ſ t® [8 (t) + 8(t + 1.8) + 8(t – 3)] dt. Express your answer using three significant figures. View Available Hint(s) ran Use step functions to write the expression for the function shown in (Figure 1)....
v e, v, z)dzdydz where f(e.v.)3 Evaluate the triple integral D and Triple Integral Region R Remember that: H(u, t, u)|J(u, v, w)ldududu F(z, y, z)dV Preview t lower limit Preview น upper limit- U lower limit Preview upper limit w lower limit upper limit H(u, o, w)- Preview Preview Ila Preview H(u, e, w)J(u,v, wdudedu Hint: The focus of this problem is on evaluating the integral and using the Jacobian. v e, v, z)dzdydz where f(e.v.)3 Evaluate the triple...
Whats the answer to number 1? 1. Let r(t) = -i-e2t j + (t? + 2t)k be the position of a particle moving in space. a. Find the particle's velocity, speed and direction at t = 0. Write the velocity as a product of speed and direction at this time. b. Find the parametric equation of the line tangent to the path of the particle at t = 0. 2. Find the integrals: a. S (tezi - 3sin(2t)j + ick)...
1 3 12. Use the transformation T: u = -x and very to evaluate the integral [JxºdA where R is the region R bounded on the xy-plane by the ellipse 9x² + 4y2 = 36. Let S be the image of R under T on the uv-plane. Sketch regions R and S. Set up the integral 7as an iterated integral of a function f(u, v) over region S. Use technology to evaluate the integral. Give the exact answer. R S...
Evaluate the line integral ∫ F *dr where C is given by the vector function r(t). F(x, y, z) = (x + y2) i + xz j + (y + z) k, r(t) = t2i + t3j − 2t k, 0 ≤ t ≤ 2
1 R 12. Use the transformation T: u = -x and very to evaluate the integral [jx?dA where R is the region bounded on the xy-plane by the ellipse 9x + 4y = 36. . Let S be the image of Runder T on the uv-plane. Sketch regions and S. Set up the integral 7as an iterated integral of a function f(u, v) over region S. Use technology to evaluate the integral. Give the exact answer. R S Y