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Chapter 11: Q. 40 (page 872)

Evaluate the integrals in Exercises 40鈥�44.

鈭�t,t2,t3dt

Short Answer

Expert verified

The value for the given vector function ist22,t33,t44+C.

Step by step solution

01

Step 1. Given Information

Given that,鈭�t,t2,t3dt.

Let us assume r(t)=t,t2,t3

02

Step 2. Evaluating the given integral

Evaluation,

鈭�r(t)dt=鈭�t,t2,t3dt=鈭�(ti+t2j+t3k)dt=i鈭�tdt+j鈭�t2dt+k鈭�t3dt=i(t22)+j(t33)+k(t44)+C=t22,t33,t44+C

So, the value of the given vector function ist22,t33,t44+C.

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Most popular questions from this chapter

Find parametric equations for each of the vector-valued functions in Exercises 26鈥�34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

r(t)=(sint,cos2t)fort鈭�[0,2蟺]

Let r(t) be a differentiable vector function. Prove that role="math" localid="1649602115972" ddt鈥�r鈥�=1鈥�r鈥�r路rr路 (Hint: role="math" localid="1649602160237" 鈥�r鈥�2=r路r

As we saw in Example 1, the graph of the vector-valued function r(t)=cost,sint,t,fort鈭�[0,2蟺] is a circular helix that spirals counterclockwise around the z-axis and climbs as t increases. Find another parametrization for this helix so that the motion is downwards.

Given a twice-differentiable vector-valued function r(t), what is the definition of the binormal vector B(t)?

Let f(t)be a differentiable scalar function and r(t)be a differentiable vector function. Prove that ddt(f(t)r(t))=f'(t)r(t)+f(t)r'(t). (This is Theorem 11.11 (b).)
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