The value (up to two decimal places) of a line integral \int_{c} F(r) \cdot d r, for \overrightarrow{F}(\overrightarrow{r})=x^{2} \hat{i}+y^{2} \hat{j} along {C} which is a straight line joining (0,0) to (1,1) is

GATE Civil Engineering (CE) 2018 Shift 2 Solved Paper

Question : 47 of 65

Marks: +1, -0

The value (up to two decimal places) of a line integral

∫↙{c} F(r) ⋅ d r, \;\;

for

{F}↖{→}({r}↖{→})=x^{2} {i}↖{∧}+y^{2} {j}↖{∧}

along

{C}

which is a straight line joining

(0,0)

(1,1)

Solution:

\ov{F} \;\; =x^{2} \ov{i}+y^{2} \ov{j}

∫ \ov{F} ⋅ d \ov{r} \;\; =∫(x^{2} \ov{i}+y^{2} \ov{j}) ⋅(d x \ov{i}+d y \ov{j})

\;\; =∫ x^{2} d x+y^{2} d y

(0,0)

(1,1)

line is

y=x

\;\; =∫ x^{2} d x+x^{2} d x=∫↙{0}↖{1} 2 x^{2} d x

\;\; =2(\;{x^{3}}/{3})_{0}^{1}=\;{2}/{3}=0.666

Go to Question:

Prev Question

Next Question