Exercise (b) the centroid of the region bounded by the graphs of the equations Step 1 The region is bounded by the graphs of the equations y = sin x, y = 0, x = 0, and x = x. y=sin(x) b $ ax The area of the representative rectangle is dA = (sin x So, the area of the entire region is A = For the above region, f(x)=sin x g(x) = 0 a = 0,b= Therefore, Let Use integration by parts. Let u= -cos x Step 2 The x-coordinate of the centroid for a region of constant density is x=x[rx) - 9(x)] dx. V= Using integration by parts Therefore, Sudv=uv - [vdu. Differentiate with respect to x on both sides. du - dx * - ** x= ✓ dv = sin x dx. Integrate with respect to x on both sides. x sin x dx --x テー ✓ sin (r) dx. -x cos x + Isin x dx -cos(r) 2 2 -[-x cos x + * cos x dx] x sin x dx. 1] The x-coordinate of the centroid of the region is

Match the set of curves with the appropriate function.

Look at the photos, all questions answered please

Rectangular (i) x=r cos 0 Polar Conversion Formulas (ii) y=r sin 0 (iii) x² + y² = r² 1. (a) Use implicit differentiation to find the slope and inclination of the tangent lines to the graph of the cardioid: (x² + y² + y)² = x² + y² at the points P = (1,0) and Q = (-1,0). (c) (b) Use the conversion formulas (i), (ii) and (iii) noted above to rewrite the equation for the cardioid from part (a) in terms of only r and as: r = 1 - sin 0. Sketch a careful graph which displays the cardioid and its tangent lines at P and Q. You should check your plot with a graphing calculator or else use the website https://www.desmos.com/calculator 2. (a) Use implicit differentiation to find the slope and inclination of the tangent line to the graph of the circle: (x - 1)² + y² = 1 at the point P = ( 1 √3 , ). 2 (b) Apply the conversion formulas (i), (ii) and (iii) to the equation from part (a) and simplify it, to obtain the polar equation of the circle: r = 2 cos 0. (c) Sketch a careful graph which displays…