Question

Consider the scalar-valued function f(x,y)=√(8y+1), and the curve C described by y=2x^2 for x∈[0,1]. The value of ∫C f(x,y)ds

is
Select one:
a. 16
b. 8
c. 19/3
d. 16/3
e. 3/32
f. 32

Answer 1

Parametric equation of curve $C$ is $r(t)=<t,2t^2>,0\leq t \leq1$

Here, differential length $ds$ is,

$ds=∥r'(t) ∥dt$

$=∥ <1,4t>∥dt$

$=\sqrt{1^2+(4t)^2}dt$

$=\sqrt{1+16t^2}dt$

Now, the line integral is evaluated as,

$\int_Cf(x,y)ds=\int_C\sqrt{8y+1}ds$

$=\int_0^{1}\sqrt{8(2t^2)+1}\sqrt{1+16t^2}dt$

$=\int_0^{1}\sqrt{1+16t^2}\sqrt{1+16t^2}dt$

$=\int_0^{1}(1+16t^2)dt$

$=[ t+16(\frac{t^3}{3})]_0^{1}$

$=1+\frac{16}{3}$

$=\frac{19}{3}$

Therefore, the line integral is $\int_Cf(x,y)ds=\frac{19}{3}$.

Hence, option (c) is correct.