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1 3. (8 points) Evaluate the contour integral (2²-22) dz 2. where C is the counterclockwise oriented curve |z-1|-

Fig: 1


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The base of a solid is the region in the xy-plane bounded by the curves * = - y² + 4y + 96 and x = y² − 22y + 140. Every cross-section of this solid perpendicular to the y-axis (and to the xy-plane) is a half-disk with the diameter of the half-disk sitting in the xy-plane. The volume of this solid is:


The base of a solid is the region in the xy-plane bounded by 4 2 -² and y = 0. Every cross-section of the curves y = 4 - 25 the solid parallel to the x-axis is a triangle whose height and base are equal. The volume of this solid is:


9.1 Add it up (#integration) For each of the following sums: (i) Estimate the sum for n= 10, 100 and 1000 (you may use computational tools to help, make sure to include supporting code). (ii) Evaluate (or estimate) the limit as n → ∞. (iii) Rewrite the sum as a definite integral and compute it. Compare your results. 2n (a) › Σ ((¹ + :-)²) = (b) Σ (COS ()) 2n


The base of a solid is the region in the xy-plane between the the lines y = x, y = 6x, x = 1 and x = 3. Cross-sections of the solid perpendicular to the x-axis (and to the xy-plane) are squares. The volume of this solid is:


A hole in the ground in the shape of an inverted cone is 22 meters deep and has radius at the top of 18 meters. This cone is filled to the top with sawdust. The density, p, of the sawdust in the hole depends upon its depth, x: p(x) = 2.7 - 1.4e-0.32 kg m³ The total mass of sawdust in the conical hole is: kg


The base of a solid is the region in the xy-plane bounded by the curves y = e² 2.2x, y = = 2.2x² +0.8 and x = 1. Every cross-section of the solid perpendicular to the x-axis (and to the xy-plane) is a square. The volume of this object is: Feel free to use technology to evaluate the integral.


(5) (5 marks) Let H be a solid hemisphere of radius a with constant density. (a) Find the centroid of H. (b) Find the moment of inertia of H about a diameter of its base.


(a) Consider the shaded area in the diagram below. i) By using integrals, find an expression for the shaded area. ii) Hence find the exact geometric area. (b) The volume of the solid formed by rotating the graph of y = f(x) about the x-axis between x = a an x = b is given by ii) Using the formula above, calculate the volume of the solid found in part(i). iii) Verify your result to part (ii) using a well-known formula. \int_{a}^{b} \pi[f(x)]^{2} d x \text { i) Describe the solid formed when the curve } y=\sqrt{64-x^{2}} \text { is rotated about } the x-axis.


\text { Set up the triple integral } \iiint_{Q} f(x, y, z) d V \text { in cylindrical coordinates. } Q \text { is the region above } z=\sqrt{x^{2}+y^{2}} \text { and below } z=\sqrt{1352-x^{2}-y^{2}} . \int_{0}^{2 \pi} \int_{0}^{26} \int_{r}^{\sqrt{1352-r^{2}}} f(r \cos (\theta), r \sin (\theta), z) d z d r d \theta \int_{0}^{2 \pi} \int_{0}^{26} \int_{r}^{\sqrt{1352-r^{2}}} f(r \cos (\theta), r \sin (\theta), z) \cdot r d z d r d \theta \int_{0}^{2 \pi} \int_{0}^{676} \int_{r}^{\sqrt{1352-r^{2}}} f(r \cos (\theta), r \sin (\theta), z) d z d r d \theta \int_{0}^{2 \pi} \int_{0}^{676} \int_{\nu}^{\sqrt{1352-r^{2}}} f(r \cos (\theta), r \sin (\theta), z) \cdot r d z d r d \theta


The point of this question is to make sure you fully understand the logic behind volumes by slicing (and aren't just plugging it into a formula that doesn't really make any sense to you.) The way to fully understand them is to understand the picture of a slice. The base of a solid is the region in the xy-plane between the the lines y = 0, y = √√x, x= 1 and 2 = 3125. = Cross-sections of the solid perpendicular to the x-axis (and to the xy-plane) are semicircles whose diameter is on the base. (So be careful with the radius of your slices.) To get full credit for this question, please do the following: 1. Draw a picture of the full base, labeling all relevant points. (This might include needing to do some algebra to determine those relevant points. Please NEATLY show that algebra work.) 2. In that base that you drew in part 1, draw a rectangle that would represent the base of one general slice and label its width and length with appropriate variable(s). 3. Draw the picture of the slice determined by that rectangle you drew in part 2. Then indicate clearly what the volume of that slice would be (In other words, what would AV be for that slice?) Note: this volume would include variable(s). 4. Lastly, write down the integral that represents the volume of the full solid and evaluate it with technology.