Sketch the region bounded by the x-axis, the curve y = x2 + x + 1 and the lines x = 2 and x = 5. Write down a definite integral that represents the area of that region
Just stuck getting points for the first equation as when I put y=0 I got a complex root?
The two vertical lines and the parabola do not bound a closed region. I suspect a typo.
Huh? Why are you doing roots when you are asked to do an integral?
the roots of x^2+ 1 x + 1 = 0 are indeed complex
now the actual problem is :
integrate y = (x^2 + x + 1) dx from x = 2 to x = 3
x^3/3 + x^2/2 + x
at 3 - at 2
= 27/3 + 9/2 + 3 - 8/3 -4/2 - 2
sorry, x = 2 to x = 5
To sketch the region bounded by the x-axis, the curve y = x^2 + x + 1, and the lines x = 2 and x = 5, we need to find the points where the curve intersects the x-axis.
To do this, we will set y = 0 in the equation y = x^2 + x + 1 and solve for x:
0 = x^2 + x + 1
This quadratic equation does not have real roots. The discriminant (b^2 - 4ac) is negative, therefore, the equation does not intersect the x-axis.
However, we can still sketch the region by considering the behavior of the equation.
The coefficient of the x^2 term is positive, indicating that the parabola opens upward. Additionally, the constant term (1) is positive, which means the vertex of the parabola is above the x-axis. Therefore, the curve y = x^2 + x + 1 does not intersect the x-axis, but it does curve upwards.
Given x = 2 and x = 5 as the boundary lines, we can still calculate the area of the region using definite integration.
To find the area of the bounded region, we need to integrate the absolute value of the curve over the given interval. Since the curve is above the x-axis for the entire interval [2, 5], we can find the area using the definite integral from x = 2 to x = 5:
Area = ∫[2,5] |x^2 + x + 1| dx
This definite integral ensures that we are calculating the area between the curve and the x-axis within the given interval.
Please note that this approach accounts for the area above the x-axis only. If you need to calculate the total area, including both above and below the x-axis, you would need to break the interval into multiple regions and integrate separately.