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Nathan Y.

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Applied Mathematics

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Question:

Approximate the following integral using Simpson's Rule with n = 8 equally spaced elements, and compare to the exact solution of the definite integral: $$ \int_0^4 \frac{1}{\sqrt{1+x^3}} dx $$

Nathan Y.

Answer:

To approximate integrals using Simpson's Rule, there must be n-evenly spaced elements with which to approximate. Simpson's Rule will not work with odd elements, because both the second and second to last elements must be multiplied by 4, and the values in between must alternate values of 2 and 4. The general formula for Simpson's Rule is: $$ \int_a^b f(x) dx \approx \frac{h}{3}[f(x_0)+4f(x_1)+2f(x_2)+...+4f(x_{n-1})+f(x_n)] $$ Where h is our step size of elements within our interval: $$ h = \frac{b-a}{n} $$ For our problem, we know that $$ a = 0, b = 4, n = 8, f(x) = \frac{1}{\sqrt{1+x^3}} $$ and we can find $$ h = \frac{4-0}{8} = \frac{1}{2} $$. We can also see that the first point, $$ x_0 = a $$, and the last point, $$ x_8 = b $$, and that the points in between $$ x_0 $$ and $$ x_8 $$ increase by increments of $$ h$$, such that $$ x_1 = a + h, x_2 = a + 2h, etc. $$ We now have enough information to set up the Simpson's Rule approximation of $$f(x)$$: $$ \int_0^4 \frac{1}{\sqrt{1+x^3}} dx \approx \frac{1/2}{3}[f(0)+4f(\frac{1}{2})+2f(1)+4f(\frac{3}{2})+2f(2)+4f(\frac{5}{2})+2f(3)+4f(\frac{7}{2})+f(4)] $$ Plugging in the value of x for each interval of f(x), we get: $$ \int_0^4 \frac{1}{\sqrt{1+x^3}} dx \approx 1.8086 $$ Which is reasonably close to the solution to the definite integral: $$ \int_0^4 \frac{1}{\sqrt{1+x^3}} dx = 1.8055 $$, giving us an error of approximately $$3.1*10^{-3}$$.

Geology

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Question:

Explain the differences between igneous, sedimentary, and metamorphic rocks.

Nathan Y.

Answer:

Igneous rocks are typically found in areas that were once magma chambers and are formed from the cooling or solidification of magma and lava. Note that magma and lava are largely the same, and the main differences are that magma is formed under the Earth's surface, while lava is magma that is exposed to the atmosphere. Sedimentary rocks are rocks that are formed by the deposition and cementation of rocks that are on the surface of the Earth. These rocks are deposited and transferred by processes of weathering and erosion through water, wind, and glaciation processes. Metamorphic rocks are the result of transforming pre-existing rocks. The pre-existing rocks in areas like fault zones get subjected to extremely high pressures and temperatures and as a result undergo large physical and chemical changes.

Mechanical Engineering

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Question:

List 3 different defects that could occur in a sand casting, and explain a good way to prevent that defect.

Nathan Y.

Answer:

1) One common defect that occurs in sand casting are inclusions, which are pockets of air bubbles that appear throughout the casted part. To prevent inclusions, there should be an effort to avoid exposure to the atmosphere, and to avoid turbulent flow or loose sand in gating system. 2) Another common defect that occurs in sand casting occurs with porosity. Because sand is a very porous material, many problems can occur with the casted part if not prepared carefully. To prevent problems involving porosity, the sand casting should be vented properly to facilitate air removal, and dissolved gases should be avoided. 3) One last common defect in sand casting occurs from shrinkage. Shrinkage happens to the casted material if there is an imbalance in the control of heat transfer, causing some parts to cool faster than other parts. To prevent this problem, risers should be added to the sand mold as necessary to control heat transfer and cast parts should be uniform in their thickness throughout the entire part.

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