Answer
See proof
Work Step by Step
Step 1: We are considering a region \(R\) that is bounded by the continuous functions \(f(x)\) and \(g(x)\) and vertical lines \(x=a\) and \(x=b\), where \(g(x) \leq f(x)\). We want to apply Green's Theorem to evaluate: \[ \int_{C} (-y) \, dx \] Where \(C\) is the boundary of the region \(R\).
Step 2: Let's state Green's Theorem: \[ \oint_{C} (f(x, y) \, dx + g(x, y) \, dy) = \iint_R \left(\frac{\partial g}{\partial x} - \frac{\partial f}{\partial y}\right) \, dA \] So, the theorem states that we can evaluate the line integral over a closed path \(C\) by evaluating the double integral on the right-hand side over the region \(R\) that is bounded by \(C\). We want to apply this theorem to evaluate the given integral.
Step 3: By comparing the left-hand side of the theorem with the given integral, we can recognize that the components in the theorem should be: \[ f(x, y) = -y, \quad g(x, y) = 0 \] The corresponding partial derivatives in the right-hand side of the theorem for these components are: \[ \frac{\partial g}{\partial x} = 0, \quad \frac{\partial f}{\partial y} = -1 \]
Step 4: Applying the above components and partial derivatives into the theorem, we get: \[ \oint_{C} (-y) \, dx = \iint_R (-1 - 0) \, dA = \iint_R (-1) \, dA \] So, the given integral is equal to the area of the region \(R\) between the curves \(f(x)\) and \(g(x)\) from \(x=a\) to \(x=b\).
Step 5: To get a more familiar formula for the area between two curves, we can split the curve \(C\) into parts \(C_1\), \(C_2\), \(C_3\), and \(C_4\), each corresponding to different parts of the boundary: \(f(x)\), \(g(x)\), \(x=a\), and \(x=b\). We can use the property of additivity for line integrals, meaning that if \(C\) is made out of smooth pieces \(C_1\), \(C_2\), etc., the integral can be split into integrals over those pieces. Using this in the equation, we get: \[ \int_{C_1} (-y) \, dx + \int_{C_2} (-y) \, dx + \int_{C_3} (-y) \, dx + \int_{C_4} (-y) \, dx = \iint_R dA \]
Step 6: Since the curves \(C_3\) and \(C_4\) are straight lines where \(x\) does not change (\(x=a\), \(x=b\)), \(\frac{d}{dx}\) is equal to 0 in these line integrals, so the whole integrals are 0. Applying this into the expression above, we have: \[ \int_{C_1} (-y) \, dx + \int_{C_2} (-y) \, dx = \iint_R dA \]
Step 7: Let us now apply the boundaries \(x=a\) and \(x=b\) into the equation: \[ \iint_R dA = \int_a^b (f(x) - g(x)) \, dx \] So, we have obtained the usual formula for finding the area between the two curves \(f(x)\) and \(g(x)\) where \(g(x) \leq f(x)\).
Step 8: To summarize, we used Green's Theorem to show that the line integral over a closed curve \(C\) is equal to the area of the region \(R\) bounded by \(C\). We divided curve \(C\) into parts corresponding to different boundaries and applied additivity to obtain the standard formula for finding the area between two curves.
