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Area under a curve Suppose the function y = h(x) is nonnegative and continuous on [α, β], which implies that the area bounded by the graph of h and x-axis on [α, β] equals
104. Show that the area of the region bounded by the ellipse x = 3 cos t, y = 4 sin t, for 0 ≤ t ≤ 2π, equals
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- Write parametric equations for a cycloid traced by a point P on a circle of radius a as the circle rolls along the x -axis given that P is at a maximum when x=0.arrow_forwardDistinguish briefly the Parametric and non-parametric analysisarrow_forwardEvaluate ∫_C▒〖(x-y)dx+(y-z)dy+xdz〗 where C is the curve from (1,-2,3) to (-4,5,2) if C consists of three line segments, the first parallel to the z-axis, the second parallel to the x-axis and the third parallel to the y-axis C is the line segmentarrow_forward
- Suppose parametric equations for the line segment between (2,4) and (−2,2) have the form:x(t)=a+bt y(t)=c+dtIf the parametric curve starts at (2,4) when t=0 and ends at (−2,2) at t=1, then find a, b, c, and d.arrow_forwardWrite parametric equations for a(t) at (1,1) a(t) = <1, t, 1/(e+t^3)>arrow_forwardThe parametric equation of the x_axis is t=1 O x=1 O x=0 O x=tarrow_forward
- The parametric equations for the line tangent to curve of intersection of the surfaces x? + y? = 8,x + y? -z = 0 at the point (2,2,8) is X-. ., y and z is known.arrow_forwardIt can be shown that the parametric equations x = x1 + (x2 – x1)t, y = y1 + (y2 – yı)t, where 0arrow_forwardDIFFERENTIATION OF PARAMETRIC EQUATIONS d²y 1.) of y = sinInß and x = In(Inß) dx2 и 2.) dx2 of y = costu – sin*u and x = 2cos? 2 3.) of s = and t = k-1 k+1 d?x 4.) of x = (02 – 1)2 and y = 403 d²y 5.) dx2 of x = 1- Int and y t-Int %D 2arrow_forwardA curve Cis given by the parametric equations z= 2t –t2 and y = (t- 2)* wheret > 0. 1. The curve has a vertical tangent line at the point O(0,-2) O(1,0) O(0, -1) O(0, 4) 2. Is the point (-8, 8) lies on the curve C? O(0,0) O(1,-1) OYes ONOarrow_forwardPartially differentiate with respect to x y and zarrow_forward_1. The Cartesian equation of the curve given in parametric equations x = e' – e-t , y = e' + e¬t is (a) x² – y² = 4. (c) x² – y² = 1. (d) y² – x² = 1. (b) y² – x² = 4. _2. The equation of the tangent line to the curve x = -2t² + 3, y = t³ – 8t at the point where t = -2 is (a) x – 2y +7 = 0. (b) x + 2y – 7 = 0. (c) 2x – y + 7 = 0. (d) None of these. _3. The slope m of the tangent line at any point on the curve r = 2 cos 0 is (b) – cot 20. (a) – tan 20. (c) tan 20. (d) None of these. _4. The graph of the equation r² = 8 cos20 is a (a) limacon (c) circle (b) cardioid. (d) lemniscate. _5. The distance between the points A(-2, –3,1) and B(6,9,-3) is (b) 2/14 units . (a) 4v14 units . (c) 8/14 units . (d) 4/7 units.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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