[amp_mcq option1=”\\[\\frac{\\pi }{2} – 1\\]” option2=”\\[\\frac{\\pi }{2} + 1\\]” option3=”\\[\\frac{\\pi }{2}\\]” option4=”1″ correct=”option1″]<\/p>\n
The correct answer is $\\boxed{\\frac{\\pi}{2}}$.<\/p>\n
The integral of a function $f(x, y)$ over a path $C$ is given by<\/p>\n
$$\\int_C f(x, y) \\, ds = \\int_a^b f(x(t), y(t)) |x'(t) + y'(t)| \\, dt$$<\/p>\n
where $(x(t), y(t))$ is a parametrization of $C$.<\/p>\n
In this case, the path $C$ is the quarter-circle $x^2 + y^2 = 1$, $y \\geq 0$, traversed in a counterclockwise sense. A parametrization of this path is<\/p>\n
$$x(t) = \\cos t, \\quad y(t) = \\sin t, \\quad 0 \\leq t \\leq \\frac{\\pi}{2}$$<\/p>\n
Substituting this into the formula for the integral, we get<\/p>\n
$$\\int_C (x + y)^2 \\, ds = \\int_0^{\\frac{\\pi}{2}} (\\cos t + \\sin t)^2 |-\\sin t + \\cos t| \\, dt = \\int_0^{\\frac{\\pi}{2}} 2 \\cos^2 t + 2 \\sin^2 t \\, dt = \\int_0^{\\frac{\\pi}{2}} 1 \\, dt = \\frac{\\pi}{2}$$<\/p>\n
Therefore, the integral of $(x + y)^2$ on path $AB$ traversed in a counterclockwise sense is $\\boxed{\\frac{\\pi}{2}}$.<\/p>\n
The other options are incorrect because they do not take into account the correct parametrization of the path $C$.<\/p>\n","protected":false},"excerpt":{"rendered":"
[amp_mcq option1=”\\[\\frac{\\pi }{2} – 1\\]” option2=”\\[\\frac{\\pi }{2} + 1\\]” option3=”\\[\\frac{\\pi }{2}\\]” option4=”1″ correct=”option1″]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[690],"tags":[],"class_list":["post-20235","post","type-post","status-publish","format-standard","hentry","category-calculus","no-featured-image-padding"],"yoast_head":"\n