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Country Boy said:So "Do my homework for me"? No, that is not going to happen! (You don't even say "please"!)
Have you graphed these so you can see what region you are working with and what the limits of integration should be? The first problem has one boundary $x^2+ y= 4$. That is the same as $y= 4- x^2$, a parabola. its "vertex" is at (0, 4) and it crosses the x-axis at (2, 0) and (-2, 0). The next is y+ x- 2= 0 which is the same as y= 2- x. That's a straight line. When x= 0, y= 2- 0= 2 and when y= 0, 2- x= 0 so x= 2. Draw the straight line through (0, 2) and (2, 0). Do you see that the line crosses the parabola where $y= 4- x^2= 2- x$. $x^2- x- 2= (x- 2)(x+ 1)= 0$ so the line crosses the parabola at (-1, 3) and (2, 0). The other two boundaries are the vertical lines x= -2 and x= 3.
Frankly that makes no sense at all! The last two vertical lines are completely outside the region bounded by the first two so there is NO region bounded by these four. If I were forced to give a numeric answer I would give the area of the region bounded by $y= 4- x^2$ and $y= 2- x$. That region goes from x= -1 on the left to x= 2 on the right and, for each x. from y= 2- x below to $y= 4- x^2$.
Imagine dividing that region into thin vertical strips of width "dx". The length of each strip is $4- x^2- (2- x)= 2+ x- x^2$ so each strip has area $(2+ x- x^2)dx$. integrate that from x= -1 to x= 2.
The purpose of studying Exam Integrals: volume and area is to understand and calculate the volume and area of various shapes and objects. This is important in many fields of science and engineering, such as physics, chemistry, and architecture.
The basic concepts of Exam Integrals: volume and area include understanding the difference between volume and area, knowing how to calculate volume and area using different methods (such as integration), and being familiar with common shapes and their corresponding formulas for volume and area.
Exam Integrals: volume and area have many real-world applications, such as calculating the volume of a swimming pool, determining the surface area of a building, and finding the volume of a chemical solution. These concepts are also used in fields like construction, manufacturing, and environmental science.
Some tips for studying Exam Integrals: volume and area include practicing solving problems regularly, understanding the underlying concepts instead of just memorizing formulas, and seeking help from a tutor or teacher if needed. It is also important to review and understand the steps involved in solving problems.
Some common mistakes to avoid when working with Exam Integrals: volume and area include using the wrong formula for a given shape, forgetting to convert units of measurement, and making calculation errors. It is also important to pay attention to the limits of integration and properly interpret the results of the problem.