Definite integrals related to geometric meaning

f(x) is greater than 0 on [a,b], strictly monotonically decreasing, convex.

s1 is the area of the graph enclosed by f(x) and x-axis.

S2 is the area of a rectangular shape with a length and width of f(b), (b-a).

S3 is the right-angled trapezoidal area that connects and extends (b,f(b)),x,f(x)) to x=a.

It's easy to see when drawn.

s1>s2,s3>s2

I can only get this result, I think s2 is the minimum.

Multiple-choice questions are good with the special value method.

Let y=x 2, which is the square of x.

a=-1, b=0-, is a very small positive number.

s1 is the area of [-1,0] at y=x 2.

f(b)=0,s2=0

s3. x 2+0) 2*(0-(-1)) x 2 2.

Obviously smaller than the S1.

Therefore, the answer is: s2, if you are not sure, you can try again with y=(x-1) 2+1.

Normal do with Taylor.

I remember I did this question, but the s3 of the original question was not like this. The original question should be:

s2〈s1〈s3。

The curve should be a positive function of monotonically decreasing. And it's concave.

S2 represents the area of a long rectangular area of the function values f(b) and (b-a) of point b. S3 represents the upper bottom with the function value of point B, the value of the function at point A is the bottom bottom, the area of the trapezoid on the x-axis (b-a) is the height, and S1 represents the area of the curved trapezoid.

It's clear s2 s1 s3.

According to yours is obviously the smallest, while S1 and S3 are not comparable. It may be S1 big or S3 big.

Student, you have a serious bias in your understanding of the median value theorem.

You go back and take a hard look at the median value theorem.

The median value theorem states that there is a point within the integral interval such that the value of the function of that point multiplied by the length of the interval is exactly the integral value.

It is not the point corresponding to the median of the range, nor the midpoint of the interval!!

"Medium" means that there is a point in the middle, not a midpoint.

Positive, decreasing convex function (image convex downward).

For example, y=1 x is such a function.

Knowing this, it's easy to know that the answer is.

。No right answer...

There is no solution, only S2 can come out to be the smallest, and S1 and S3 cannot be judged.

The main idea is to draw an image according to the upper and lower limits of the integrand and variables, and use the geometric meaning of the integral segment to solve.

The solution is as follows: grasp the seepage diagram of the god worship section of the blind spine.

The following solution is better understood in combination with the function of the graph base volt image.

1) The value of this integral is the area of an isosceles right triangle with a right angle side of 2, so the answer is 2*2*

2) The value of this integral is a quarter of the area of a circle with radius a, so the answer is 3) sinx is a periodic function with a period of 2pi, and it is an odd function, so its integral value on [-pi, 0] and [0, pi] is the same as the Hui calendar, and the sign is opposite, so the answer is 0

The definite integral is an important concept in calculus. The German mathematician Riemann first gave a strict expression, so it is also known as the "Rietuan Chaman integral". As we all know, the two major parts of calculus are differentiation and integration.

In the case of a univariate function, finding the differential is actually finding the derivative of a known function, while the integral is the derivative of a known function, finding the original Lawang function. Therefore, differentiation and integration are mutually reversible.

Chinese name fixed points.

Let y= (2x-x), so y = 2x-x, i.e. (x-1) +y =1

The geometric meaning of the dividing world is not discussed: (0,2) Wide limb (2x-x) dx= *1 2= 2

According to the geometric meaning of the definite integral, problem 1 represents the area of the upper semicircle of a circle with the origin (0,0) as the center of the circle and radius r=a, and its value is (1 2) r = a 2.

Question 2 shows the area enclosed by y=sinx and x-axis when x [0,2 ], when x [0,2 ], is expressed. At x [0, ] is a positive value, and at x [ 2 ], a negative value. And the absolute values of the positive and negative values are equal. Original = 0.

FYI.

y=√(4-x²)>0

x sock ball + y = 4

So this is a macro semicircle above the x-axis.

0<=x<=2

So it's just the first quadrant.

So it's an area of 1 4 circles.

The radius is 2 so 1 4 circle area =

So definite integral =

In the buried field (1, 3) with trouble.

5x-2>0 so.

Its geometric meaning is to bend and shout yes. In order to. x=1

x=3y=0

y=5x-2 The area of the trapezoid, which consists of four sides, i.e.

At. Above.

When we already know it, definite integrals.

Geometrically represents curves.

Two straight lines. And. the area of the curved trapezoidal enclosed by the shaft; At.

Above. time, by the curve.

Two straight lines. And. The curved trapezoidal shape enclosed by the shaft is located.

Below the axis, the definite integral.

Geometrically represents the negative value of the above-mentioned curved trapezoidal area; At.

Above. When both positive and negative values are obtained, the function.

Some parts of the graph are in.

axis above, while the other parts are in.

Underneath the axis. If we give a plus or minus sign to the area, in.

The area of the graph above the axis is assigned a positive sign, in.

The area of the graph below the axis is assigned a negative sign, and in general, the integral is determined.

The geometric meaning is: it is between.

axes, functions. and two curves.

The algebraic sum between the areas of each part.

y = 1-x squared under the root number.

This image is the part above the x-axis of the unit circle.

Based on your answer, I'm guessing the integral limit is [0,1].

The integral in this interval is the area of 1 4 unit circle.

So it's 4

High-quality answers.

The first one: cosx (see the figure) is symmetrical with respect to x=pi, one side is above the x-axis and the other is below the x-axis, and the areas cancel each other out, so it is 0

The second: y= a 2-x 2 is a 1 4 circle, and the definite integral calculates the area, whose area is pai*a 2 4.

In (1,3), 5x-2>0, so its geometric meaning is the area of a trapezoid with four sides of x=1, x=3, y=0, y=5x-2, i.e., (3+13) 2, 2=16

The geometric meaning is the area of a trapezoid with four sides of x=1, x=3, y=0, y=5x-2, i.e., (3+13) 2, 2=16 jgalkj